Sequences

392,541 sequences

sech(arctan(arctan(x)))=1-1/2!*x^2+21/4!*x^4-1149/6!*x^6+119305/8!*x^8...A012211
sech(arctan(arctan(x)))=1-1/2!*x^2+21/4!*x^4-1149/6!*x^6+119305/8!*x^8...
Number of squarefree palindromes over {0, 1, 2} of length 2n+1.A012212
Number of squarefree palindromes over {0, 1, 2} of length 2n+1.
exp(arctan(arcsinh(x)))=1+x+1/2!*x^2-2/3!*x^3-11/4!*x^4+24/5!*x^5...A012213
exp(arctan(arcsinh(x)))=1+x+1/2!*x^2-2/3!*x^3-11/4!*x^4+24/5!*x^5...
arcsin(arctan(arcsinh(x)))=x-2/3!*x^3+32/5!*x^5-1280/7!*x^7+96896/9!*x^9...A012214
arcsin(arctan(arcsinh(x)))=x-2/3!*x^3+32/5!*x^5-1280/7!*x^7+96896/9!*x^9...
arctan(arctan(arcsinh(x)))=x-5/3!*x^3+137/5!*x^5-9029/7!*x^7...A012215
arctan(arctan(arcsinh(x)))=x-5/3!*x^3+137/5!*x^5-9029/7!*x^7...
sinh(arctan(arcsinh(x)))=x-2/3!*x^3+24/5!*x^5-664/7!*x^7+28416/9!*x^9...A012216
sinh(arctan(arcsinh(x)))=x-2/3!*x^3+24/5!*x^5-664/7!*x^7+28416/9!*x^9...
arcsinh(arctan(arcsinh(x)))=x-4/3!*x^3+92/5!*x^5-5216/7!*x^7...A012217
arcsinh(arctan(arcsinh(x)))=x-4/3!*x^3+92/5!*x^5-5216/7!*x^7...
tanh(arctan(arcsinh(x)))=x-5/3!*x^3+129/5!*x^5-7741/7!*x^7...A012218
tanh(arctan(arcsinh(x)))=x-5/3!*x^3+129/5!*x^5-7741/7!*x^7...
arctanh(arctan(arcsinh(x)))=x-1/3!*x^3+17/5!*x^5-617/7!*x^7+44417/9!*x^9...A012219
arctanh(arctan(arcsinh(x)))=x-1/3!*x^3+17/5!*x^5-617/7!*x^7+44417/9!*x^9...
cosh(arctan(arcsinh(x)))=1+1/2!*x^2-11/4!*x^4+349/6!*x^6-22007/8!*x^8...A012220
cosh(arctan(arcsinh(x)))=1+1/2!*x^2-11/4!*x^4+349/6!*x^6-22007/8!*x^8...
sech(arctan(arcsinh(x)))=1-1/2!*x^2+17/4!*x^4-769/6!*x^6+66401/8!*x^8...A012221
sech(arctan(arcsinh(x)))=1-1/2!*x^2+17/4!*x^4-769/6!*x^6+66401/8!*x^8...
Expansion of e.g.f.: exp(arctan(tanh(x)))=1+x+1/2!*x^2-3/3!*x^3-15/4!*x^4+41/5!*x^5...A012222
Expansion of e.g.f.: exp(arctan(tanh(x)))=1+x+1/2!*x^2-3/3!*x^3-15/4!*x^4+41/5!*x^5...
arcsin(arctan(tanh(x)))=x-3/3!*x^3+49/5!*x^5-2139/7!*x^7+181921/9!*x^9...A012223
arcsin(arctan(tanh(x)))=x-3/3!*x^3+49/5!*x^5-2139/7!*x^7+181921/9!*x^9...
arctan(arctan(tanh(x)))=x-6/3!*x^3+184/5!*x^5-13584/7!*x^7...A012224
arctan(arctan(tanh(x)))=x-6/3!*x^3+184/5!*x^5-13584/7!*x^7...
sinh(arctan(tanh(x)))=x-3/3!*x^3+41/5!*x^5-1243/7!*x^7+58001/9!*x^9...A012225
sinh(arctan(tanh(x)))=x-3/3!*x^3+41/5!*x^5-1243/7!*x^7+58001/9!*x^9...
arcsinh(arctan(tanh(x)))=x-5/3!*x^3+129/5!*x^5-8189/7!*x^7...A012226
arcsinh(arctan(tanh(x)))=x-5/3!*x^3+129/5!*x^5-8189/7!*x^7...
tanh(arctan(tanh(x)))=x-6/3!*x^3+176/5!*x^5-12016/7!*x^7...A012227
tanh(arctan(tanh(x)))=x-6/3!*x^3+176/5!*x^5-12016/7!*x^7...
arctanh(arctan(tanh(x)))=x-2/3!*x^3+24/5!*x^5-944/7!*x^7+77952/9!*x^9...A012228
arctanh(arctan(tanh(x)))=x-2/3!*x^3+24/5!*x^5-944/7!*x^7+77952/9!*x^9...
cosh(arctan(tanh(x)))=1+1/2!*x^2-15/4!*x^4+561/6!*x^6-40415/8!*x^8...A012229
cosh(arctan(tanh(x)))=1+1/2!*x^2-15/4!*x^4+561/6!*x^6-40415/8!*x^8...
sech(arctan(tanh(x)))=1-1/2!*x^2+21/4!*x^4-1101/6!*x^6+109001/8!*x^8...A012230
sech(arctan(tanh(x)))=1-1/2!*x^2+21/4!*x^4-1101/6!*x^6+109001/8!*x^8...
exp(arctan(arctanh(x)))=1+x+1/2!*x^2+1/3!*x^3+1/4!*x^4+9/5!*x^5...A012231
exp(arctan(arctanh(x)))=1+x+1/2!*x^2+1/3!*x^3+1/4!*x^4+9/5!*x^5...
arcsin(arctan(arctanh(x)))=x+1/3!*x^3+17/5!*x^5+505/7!*x^7+32321/9!*x^9...A012232
arcsin(arctan(arctanh(x)))=x+1/3!*x^3+17/5!*x^5+505/7!*x^7+32321/9!*x^9...
arctan(arctan(arctanh(x)))=x-2/3!*x^3+32/5!*x^5-944/7!*x^7+64640/9!*x^9...A012233
arctan(arctan(arctanh(x)))=x-2/3!*x^3+32/5!*x^5-944/7!*x^7+64640/9!*x^9...
sinh(arctan(arctanh(x)))=x+1/3!*x^3+9/5!*x^5+281/7!*x^7+13233/9!*x^9...A012234
sinh(arctan(arctanh(x)))=x+1/3!*x^3+9/5!*x^5+281/7!*x^7+13233/9!*x^9...
E.g.f.: arcsinh(arctan(arctanh(x)))=x-1/3!*x^3+17/5!*x^5-281/7!*x^7+24257/9!*x^9...A012235
E.g.f.: arcsinh(arctan(arctanh(x)))=x-1/3!*x^3+17/5!*x^5-281/7!*x^7+24257/9!*x^9...
tanh(arctan(arctanh(x)))=x-2/3!*x^3+24/5!*x^5-496/7!*x^7+24192/9!*x^9...A012236
tanh(arctan(arctanh(x)))=x-2/3!*x^3+24/5!*x^5-496/7!*x^7+24192/9!*x^9...
arctanh(arctan(arctanh(x)))=x+2/3!*x^3+32/5!*x^5+1168/7!*x^7+80768/9!*x^9...A012237
arctanh(arctan(arctanh(x)))=x+2/3!*x^3+32/5!*x^5+1168/7!*x^7+80768/9!*x^9...
cosh(arctan(arctanh(x)))=1+1/2!*x^2+1/4!*x^4+49/6!*x^6+1345/8!*x^8...A012238
cosh(arctan(arctanh(x)))=1+1/2!*x^2+1/4!*x^4+49/6!*x^6+1345/8!*x^8...
sech(arctan(arctanh(x)))=1-1/2!*x^2+5/4!*x^4-109/6!*x^6+2729/8!*x^8...A012239
sech(arctan(arctanh(x)))=1-1/2!*x^2+5/4!*x^4-109/6!*x^6+2729/8!*x^8...
-log(cos(tan(x)))=1/2!*x^2+10/4!*x^4+232/6!*x^6+10064/8!*x^8...A012240
-log(cos(tan(x)))=1/2!*x^2+10/4!*x^4+232/6!*x^6+10064/8!*x^8...
log(cos(arcsinh(x))) = -1/2!*x^2+2/4!*x^4-40/6!*x^6+1360/8!*x^8...A012241
log(cos(arcsinh(x))) = -1/2!*x^2+2/4!*x^4-40/6!*x^6+1360/8!*x^8...
log(cos(tanh(x)))=-1/2!*x^2+6/4!*x^4-72/6!*x^6+1456/8!*x^8...A012242
log(cos(tanh(x)))=-1/2!*x^2+6/4!*x^4-72/6!*x^6+1456/8!*x^8...
-log(cos(arctanh(x)))=1/2!*x^2+10/4!*x^4+280/6!*x^6+15440/8!*x^8...A012243
-log(cos(arctanh(x)))=1/2!*x^2+10/4!*x^4+280/6!*x^6+15440/8!*x^8...
a(n+2) = (2n+3)*a(n+1) + (n+1)^2*a(n), a(0) = 1, a(1) = 1.A012244
a(n+2) = (2n+3)*a(n+1) + (n+1)^2*a(n), a(0) = 1, a(1) = 1.
Characteristic function of factorial numbers; also decimal expansion of Liouville's number or Liouville's constant.A012245
Characteristic function of factorial numbers; also decimal expansion of Liouville's number or Liouville's constant.
E.g.f.: exp(sinh(arcsin(x)))=1+x+1/2!*x^2+3/3!*x^3+9/4!*x^4+41/5!*x^5...A012246
E.g.f.: exp(sinh(arcsin(x)))=1+x+1/2!*x^2+3/3!*x^3+9/4!*x^4+41/5!*x^5...
exp(sinh(arctan(x)))=1+x+1/2!*x^2-3/4!*x^4-4/5!*x^5+21/6!*x^6...A012247
exp(sinh(arctan(x)))=1+x+1/2!*x^2-3/4!*x^4-4/5!*x^5+21/6!*x^6...
Expansion of e.g.f. exp(arcsinh(arcsin(x))).A012248
Expansion of e.g.f. exp(arcsinh(arcsin(x))).
Volume of a certain rational polytope whose points with given denominator count certain sets of Standard Tableaux.A012249
Volume of a certain rational polytope whose points with given denominator count certain sets of Standard Tableaux.
a(n) = A012249(2*n) / 2^(2*n-1).A012250
a(n) = A012249(2*n) / 2^(2*n-1).
exp(arcsinh(arctan(x)))=1+x+1/2!*x^2-2/3!*x^3-11/4!*x^4+24/5!*x^5...A012251
exp(arcsinh(arctan(x)))=1+x+1/2!*x^2-2/3!*x^3-11/4!*x^4+24/5!*x^5...
exp(arcsinh(arcsinh(x))) = 1+x+1/2!*x^2-1/3!*x^3-7/4!*x^4+9/5!*x^5...A012252
exp(arcsinh(arcsinh(x))) = 1+x+1/2!*x^2-1/3!*x^3-7/4!*x^4+9/5!*x^5...
exp(arcsinh(tanh(x))) = 1+x+1/2!*x^2-2/3!*x^3-11/4!*x^4+16/5!*x^5...A012253
exp(arcsinh(tanh(x))) = 1+x+1/2!*x^2-2/3!*x^3-11/4!*x^4+16/5!*x^5...
exp(arcsinh(arctanh(x)))=1+x+1/2!*x^2+2/3!*x^3+5/4!*x^4+24/5!*x^5...A012254
exp(arcsinh(arctanh(x)))=1+x+1/2!*x^2+2/3!*x^3+5/4!*x^4+24/5!*x^5...
Expansion of e.g.f.: exp(tanh(arcsin(x)))=1+x+1/2!*x^2-3/4!*x^4-4/5!*x^5+21/6!*x^6...A012255
Expansion of e.g.f.: exp(tanh(arcsin(x)))=1+x+1/2!*x^2-3/4!*x^4-4/5!*x^5+21/6!*x^6...
Expansion of e.g.f.: exp(tanh(arctan(x)))=1+x+1/2!*x^2-3/3!*x^3-15/4!*x^4+41/5!*x^5...A012256
Expansion of e.g.f.: exp(tanh(arctan(x)))=1+x+1/2!*x^2-3/3!*x^3-15/4!*x^4+41/5!*x^5...
Irregular triangle read by rows: row 0 is {2}; if row n is {r_1, ..., r_k} then row n+1 is {r_k 1's, r_{k-1} 2's, r_{k-2} 3's, etc.}.A012257
Irregular triangle read by rows: row 0 is {2}; if row n is {r_1, ..., r_k} then row n+1 is {r_k 1's, r_{k-1} 2's, r_{k-2} 3's, etc.}.
Expansion of e.g.f.: exp(arctanh(arcsin(x)))=1+x+1/2!*x^2+4/3!*x^3+13/4!*x^4+84/5!*x^5...A012258
Expansion of e.g.f.: exp(arctanh(arcsin(x)))=1+x+1/2!*x^2+4/3!*x^3+13/4!*x^4+84/5!*x^5...
Expansion of e.g.f. exp(arctanh(tan(x))).A012259
Expansion of e.g.f. exp(arctanh(tan(x))).
Expansion of e.g.f. exp(arctanh(arctan(x))).A012260
Expansion of e.g.f. exp(arctanh(arctan(x))).