Sequences

392,541 sequences

tanh(sec(x)-sech(x))=2/2!*x^2-118/6!*x^6-29998/10!*x^10...A013511
tanh(sec(x)-sech(x))=2/2!*x^2-118/6!*x^6-29998/10!*x^10...
arctanh(sec(x)-sech(x))=2/2!*x^2+362/6!*x^6+1441682/10!*x^10...A013512
arctanh(sec(x)-sech(x))=2/2!*x^2+362/6!*x^6+1441682/10!*x^10...
cosh(sec(x)-sech(x)) = 1+12/4!*x^4+8512/8!*x^8+34406592/12!*x^12...A013513
cosh(sec(x)-sech(x)) = 1+12/4!*x^4+8512/8!*x^8+34406592/12!*x^12...
sec(sec(x)-sech(x)) = 1+12/4!*x^4+15232/8!*x^8+128432832/12!*x^12...A013514
sec(sec(x)-sech(x)) = 1+12/4!*x^4+15232/8!*x^8+128432832/12!*x^12...
sech(sec(x)-sech(x))=1-12/4!*x^4+1568/8!*x^8+6840768/12!*x^12...A013515
sech(sec(x)-sech(x))=1-12/4!*x^4+1568/8!*x^8+6840768/12!*x^12...
Denominators in the Taylor expansion exp(cosec(x)-cot(x))=1 + x/2 + x^2/8 + x^3/16 + 3*x^4/128 + 37*x^5/3840 + 59*x^6/15360 + ...A013516
Denominators in the Taylor expansion exp(cosec(x)-cot(x))=1 + x/2 + x^2/8 + x^3/16 + 3*x^4/128 + 37*x^5/3840 + 59*x^6/15360 + ...
Denominator of [x^(2n+1)] in the Taylor expansion sin(cosec(x)-cot(x))= x/2 + x^3/48 - x^5/1280 - 55*x^7/129024 - 143*x^9/1769472 + ...A013517
Denominator of [x^(2n+1)] in the Taylor expansion sin(cosec(x)-cot(x))= x/2 + x^3/48 - x^5/1280 - 55*x^7/129024 - 143*x^9/1769472 + ...
Numerator of [x^(2n+1)] in the Taylor expansion arcsin(cosec(x)-cot(x)) = x/2 + x^3/16 + 3*x^5/256 + 83*x^7/30720 + 8887*x^9/12386304 + ...A013518
Numerator of [x^(2n+1)] in the Taylor expansion arcsin(cosec(x)-cot(x)) = x/2 + x^3/16 + 3*x^5/256 + 83*x^7/30720 + 8887*x^9/12386304 + ...
Numerator of [x^(2n+1)] in the Taylor expansion tan(cosec(x)-cot(x)).A013519
Numerator of [x^(2n+1)] in the Taylor expansion tan(cosec(x)-cot(x)).
A problem in derangements.A013520
A problem in derangements.
Numerator of [x^(2n)] in the Taylor expansion cos(cosec(x)-cot(x))= 1-x^2/8 -7*x^4/384 -97*x^6/46080 -2063*x^8/10321920 -17803*x^10/1238630400 -....A013521
Numerator of [x^(2n)] in the Taylor expansion cos(cosec(x)-cot(x))= 1-x^2/8 -7*x^4/384 -97*x^6/46080 -2063*x^8/10321920 -17803*x^10/1238630400 -....
Numerator of [x^(2n+1)] in the Taylor expansion sinh(cosec(x)-cotan(x))= x/2 +x^3/16 +37*x^5/3840 +137*x^7/92160 +41641*x^9/185794560 + 3887*x^11/117964800 +...A013522
Numerator of [x^(2n+1)] in the Taylor expansion sinh(cosec(x)-cotan(x))= x/2 +x^3/16 +37*x^5/3840 +137*x^7/92160 +41641*x^9/185794560 + 3887*x^11/117964800 +...
Denominator of [x^(2n+1)] in the Taylor expansion arcsinh(cosec(x) - cot(x)).A013523
Denominator of [x^(2n+1)] in the Taylor expansion arcsinh(cosec(x) - cot(x)).
Numerator of [x^(2n+1)] of the Taylor expansion tanh(cosec(x) - cot(x)).A013524
Numerator of [x^(2n+1)] of the Taylor expansion tanh(cosec(x) - cot(x)).
E.g.f.: x + (gdinv x - sinh x)/2, where gdinv = inverse-Gudermannian. Sequence has odd-indexed coefficients; others are zero.A013525
E.g.f.: x + (gdinv x - sinh x)/2, where gdinv = inverse-Gudermannian. Sequence has odd-indexed coefficients; others are zero.
Numerator of [x^(2n)] of the Taylor expansion cosh(cosec(x)-cot(x))=1 +x^2/8 +3*x^4/128 +59*x^6/15360 +871*x^8/1474560 +....A013526
Numerator of [x^(2n)] of the Taylor expansion cosh(cosec(x)-cot(x))=1 +x^2/8 +3*x^4/128 +59*x^6/15360 +871*x^8/1474560 +....
Numerator of [x^(2n)] of the Taylor expansion sec(cosec(x)-cot(x)) = 1+ x^2/8 +13*x^4/384 +397*x^6/46080 +4453*x^8/2064384 + ... .A013527
Numerator of [x^(2n)] of the Taylor expansion sec(cosec(x)-cot(x)) = 1+ x^2/8 +13*x^4/384 +397*x^6/46080 +4453*x^8/2064384 + ... .
Numerator of [x^(2n)] of the Taylor series sech(cosec(x)-cot(x)) = 1 -x^2/8 -x^4/128 +x^6/15360 +19*x^8/294912 +25031*x^10/3715891200+... .A013528
Numerator of [x^(2n)] of the Taylor series sech(cosec(x)-cot(x)) = 1 -x^2/8 -x^4/128 +x^6/15360 +19*x^8/294912 +25031*x^10/3715891200+... .
Numerator of [x^n] in the Taylor expansion exp(cosec(x) - cosech(x)) = 1 + x/3 + x^2/18 + x^3/162 + x^4/1944 + 211*x^5/51030 + ...A013529
Numerator of [x^n] in the Taylor expansion exp(cosec(x) - cosech(x)) = 1 + x/3 + x^2/18 + x^3/162 + x^4/1944 + 211*x^5/51030 + ...
Numerator of [x^(2n+1)] of the Taylor expansion sin(cosec(x) - cosech(x)) = x/3 - x^3/162 + 211*x^5/51030 -157*x^7/688905 + 439801*x^9/9821029680 - ...A013530
Numerator of [x^(2n+1)] of the Taylor expansion sin(cosec(x) - cosech(x)) = x/3 - x^3/162 + 211*x^5/51030 -157*x^7/688905 + 439801*x^9/9821029680 - ...
Numerator of [x^(2n+1)] in the Taylor expansion arcsin(cosec(x) - cosech(x)) = x/3 + x^3/162 + 5*x^5/1134 + 19*x^7/76545 + 13793*x^9/218245104 + ...A013531
Numerator of [x^(2n+1)] in the Taylor expansion arcsin(cosec(x) - cosech(x)) = x/3 + x^3/162 + 5*x^5/1134 + 19*x^7/76545 + 13793*x^9/218245104 + ...
Numerator of [x^(2n+1)] in the Taylor series tan(cosec(x)-cosech(x)) = x/3 +x^3/81 +949*x^5/204120 +2647*x^7/5511240+... .A013532
Numerator of [x^(2n+1)] in the Taylor series tan(cosec(x)-cosech(x)) = x/3 +x^3/81 +949*x^5/204120 +2647*x^7/5511240+... .
Numerator of [x^(2n+1)] in the Taylor expansion arctan(cosec(x) - cosech(x)).A013533
Numerator of [x^(2n+1)] in the Taylor expansion arctan(cosec(x) - cosech(x)).
Numerator of [x^(2n)] in the Taylor expansion cos(cosec(x) - cosech(x)).A013534
Numerator of [x^(2n)] in the Taylor expansion cos(cosec(x) - cosech(x)).
Numerator of the coefficient [x^(2n)] in the Taylor series sec(cosec(x) - cosech(x)).A013535
Numerator of the coefficient [x^(2n)] in the Taylor series sec(cosec(x) - cosech(x)).
Numerators of coefficients in Taylor series expansion of exp(cosec(x)-cotanh(x)).A013536
Numerators of coefficients in Taylor series expansion of exp(cosec(x)-cotanh(x)).
Numerator of [x^(2n+1)] of the Taylor expansion sin(cosec(x)-coth(x)).A013537
Numerator of [x^(2n+1)] of the Taylor expansion sin(cosec(x)-coth(x)).
Numerator of [x^(2*n + 1)] in the Taylor series expansion of arcsin(cosec(x) - cotanh(x)).A013538
Numerator of [x^(2*n + 1)] in the Taylor series expansion of arcsin(cosec(x) - cotanh(x)).
Numerator of the coefficient [x^(2n+1)] of the Taylor series tan(cosec(x)-coth(x)).A013539
Numerator of the coefficient [x^(2n+1)] of the Taylor series tan(cosec(x)-coth(x)).
Numerator of the coefficient [x^(2n+1)] of the Taylor series arctan(cosec(x) - coth(x)).A013540
Numerator of the coefficient [x^(2n+1)] of the Taylor series arctan(cosec(x) - coth(x)).
Numerator of the coefficient [x^(2n)] of the Taylor series cos(cosec(x) - coth(x)).A013541
Numerator of the coefficient [x^(2n)] of the Taylor series cos(cosec(x) - coth(x)).
Numerator of the coefficient [x^(2n+1)] of the Taylor series sinh(cosec(x) - coth(x)).A013542
Numerator of the coefficient [x^(2n+1)] of the Taylor series sinh(cosec(x) - coth(x)).
Numerator of the coefficient [x^(2n+1)] of the Taylor series arcsinh(cosec(x) - coth(x)).A013543
Numerator of the coefficient [x^(2n+1)] of the Taylor series arcsinh(cosec(x) - coth(x)).
Numerator of the coefficient [x^(2n+1)] of tanh(cosec(x)-coth(x)).A013544
Numerator of the coefficient [x^(2n+1)] of tanh(cosec(x)-coth(x)).
Numerator of the coefficient [x^(2n+1)] of the Taylor series arctanh(cosec(x) - coth(x)).A013545
Numerator of the coefficient [x^(2n+1)] of the Taylor series arctanh(cosec(x) - coth(x)).
Numerator of [x^(2n)] in the Taylor expansion cosh(cosec(x)-coth(x))=1 +x^2/72 -215*x^4/31104 +199159*x^6/235146240-...A013546
Numerator of [x^(2n)] in the Taylor expansion cosh(cosec(x)-coth(x))=1 +x^2/72 -215*x^4/31104 +199159*x^6/235146240-...
Numerator of the coefficient [x^(2n)] of the Taylor expansion sec(cosec(x)-coth(x))= 1+x^2/72 -211*x^4/31104 +169339*x^6/235146240 -205787*x^8/13544423424+...A013547
Numerator of the coefficient [x^(2n)] of the Taylor expansion sec(cosec(x)-coth(x))= 1+x^2/72 -211*x^4/31104 +169339*x^6/235146240 -205787*x^8/13544423424+...
Numerator of [x^(2n)] of the Taylor series sech(cosec(x)-coth(x))= 1-x^2/72 +221*x^4/31104 +244939*x^6/235146240 +1846213*x^8/13544423424-...A013548
Numerator of [x^(2n)] of the Taylor series sech(cosec(x)-coth(x))= 1-x^2/72 +221*x^4/31104 +244939*x^6/235146240 +1846213*x^8/13544423424-...
Incorrect version of A000085.A013549
Incorrect version of A000085.
Numerator of the coefficient of [x^n] in the Taylor expansion exp(cot(x) - cosech(x)).A013550
Numerator of the coefficient of [x^n] in the Taylor expansion exp(cot(x) - cosech(x)).
Numerator of [x^n] in the Taylor expansion exp(cot(x)-coth(x))= 1-2*x/3 +2x^2/9 -4*x^3/81 +2*x^4/243 -136*x^5/25515 +676*x^6/229635 -...A013551
Numerator of [x^n] in the Taylor expansion exp(cot(x)-coth(x))= 1-2*x/3 +2x^2/9 -4*x^3/81 +2*x^4/243 -136*x^5/25515 +676*x^6/229635 -...
Numerator of [x^(2n+1)] of the Taylor series sin(cot(x)-coth(x)).A013552
Numerator of [x^(2n+1)] of the Taylor series sin(cot(x)-coth(x)).
Numerator of [x^(2n+1)] of the Taylor expansion -arcsin(cot(x)-coth(x)).A013553
Numerator of [x^(2n+1)] of the Taylor expansion -arcsin(cot(x)-coth(x)).
Numerator of [x^(2n+1)] in the Taylor expansion -tan(cot(x)-coth(x))= 2*x/3 +8*x^3/81 +556*x^5/25515 +496*x^7/98415 +717572*x^9/613814355 +...A013554
Numerator of [x^(2n+1)] in the Taylor expansion -tan(cot(x)-coth(x))= 2*x/3 +8*x^3/81 +556*x^5/25515 +496*x^7/98415 +717572*x^9/613814355 +...
Numerator of [x^(2n+1)] of the Taylor series arctan(cot(x)-coth(x))= -2*x/3 +8*x^3/81 -52*x^5/1701 +112*x^7/10935 -51412*x^9/13640319+...A013555
Numerator of [x^(2n+1)] of the Taylor series arctan(cot(x)-coth(x))= -2*x/3 +8*x^3/81 -52*x^5/1701 +112*x^7/10935 -51412*x^9/13640319+...
Numerator of [x^(2n)] of the Taylor series cos(cot(x)-coth(x))= 1 -2*x^2/9 +2*x^4/243 -676*x^6/229635 +62*x^8/295245 -...A013556
Numerator of [x^(2n)] of the Taylor series cos(cot(x)-coth(x))= 1 -2*x^2/9 +2*x^4/243 -676*x^6/229635 +62*x^8/295245 -...
Numerator of [x^(2n)] of the Taylor series sec(cot(x)-coth(x))= 1 +2*x^2/9 +10*x^4/243 +2356*x^6/229635 +986*x^8/413343+...A013557
Numerator of [x^(2n)] of the Taylor series sec(cot(x)-coth(x))= 1 +2*x^2/9 +10*x^4/243 +2356*x^6/229635 +986*x^8/413343+...
E.g.f. -log(sech(x) - log(x+1)).A013558
E.g.f. -log(sech(x) - log(x+1)).
Numerator of [x^n] in the Taylor expansion of exp(cosech(x)-coth(x)).A013559
Numerator of [x^n] in the Taylor expansion of exp(cosech(x)-coth(x)).
Cardinality of the permutation (k, k-1, ..., 2, 1)(n, n-1, ..., k+1) in an exchange shuffle applied in all n^n possible ways to (1,2,...,n).A013560
Cardinality of the permutation (k, k-1, ..., 2, 1)(n, n-1, ..., k+1) in an exchange shuffle applied in all n^n possible ways to (1,2,...,n).