Sequences

392,541 sequences

Expansion of 1/((1-2x)(1-3x)(1-4x)(1-5x)).A025211
Expansion of 1/((1-2x)(1-3x)(1-4x)(1-5x)).
a(n) = floor(2nd elementary symmetric function of Sum_{j=1..k} 1/j, k = 1,2,...,n).A025212
a(n) = floor(2nd elementary symmetric function of Sum_{j=1..k} 1/j, k = 1,2,...,n).
a(n) = floor(3rd elementary symmetric function of Sum_{j=1..k} 1/j, k = 1,2,...,n).A025213
a(n) = floor(3rd elementary symmetric function of Sum_{j=1..k} 1/j, k = 1,2,...,n).
a(n) = floor(4th elementary symmetric function of Sum_{j=1..k} 1/j, k = 1,2,...,n).A025214
a(n) = floor(4th elementary symmetric function of Sum_{j=1..k} 1/j, k = 1,2,...,n).
a(n) = [ Sum of squares of 1 + 1/2 + ... + 1/k ], k = 1,2,...,n.A025215
a(n) = [ Sum of squares of 1 + 1/2 + ... + 1/k ], k = 1,2,...,n.
[ Sum (s(j) - s(i))^2 ], 1 <= i < j <= n, where s(k) = 1 + 1/2 + ... + 1/k.A025216
[ Sum (s(j) - s(i))^2 ], 1 <= i < j <= n, where s(k) = 1 + 1/2 + ... + 1/k.
[ Sum (s(j) - s(i))^3 ], 1 <= i < j <= n, where s(k) = 1 + 1/2 + ... + 1/k.A025217
[ Sum (s(j) - s(i))^3 ], 1 <= i < j <= n, where s(k) = 1 + 1/2 + ... + 1/k.
a(n) = floor( Sum_{k=1..n} sqrt(k+1) ).A025218
a(n) = floor( Sum_{k=1..n} sqrt(k+1) ).
a(n) = [ 2nd elementary symmetric function of {sqrt(k+1)} ], k = 1,2,...,n.A025219
a(n) = [ 2nd elementary symmetric function of {sqrt(k+1)} ], k = 1,2,...,n.
[ 3rd elementary symmetric function of {sqrt(k+1)} ], k = 1,2,...,n.A025220
[ 3rd elementary symmetric function of {sqrt(k+1)} ], k = 1,2,...,n.
[ 4th elementary symmetric function of sqrt(k+1) ], k = 1,2,...,n.A025221
[ 4th elementary symmetric function of sqrt(k+1) ], k = 1,2,...,n.
a(n) = [ Sum{(sqrt(j+1)-sqrt(i+1))^2} ], 1 <= i < j <= n.A025222
a(n) = [ Sum{(sqrt(j+1)-sqrt(i+1))^2} ], 1 <= i < j <= n.
[ Sum{(sqrt(j+1)-sqrt(i+1))^3} ], 1 <= i < j <= n.A025223
[ Sum{(sqrt(j+1)-sqrt(i+1))^3} ], 1 <= i < j <= n.
a(n) = floor(Sum_{k=1..n} sqrt(k)).A025224
a(n) = floor(Sum_{k=1..n} sqrt(k)).
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 2. Also a(n) = (2^n)*C(n-1), where C = A000108 (Catalan numbers).A025225
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 2. Also a(n) = (2^n)*C(n-1), where C = A000108 (Catalan numbers).
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 2. Also a(n) = 3^n*C(n-1), where C = A000108 (Catalan numbers).A025226
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 2. Also a(n) = 3^n*C(n-1), where C = A000108 (Catalan numbers).
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-1)*a(1) for n >= 3.A025227
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-1)*a(1) for n >= 3.
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 3, with initial terms 2,1.A025228
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 3, with initial terms 2,1.
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 3, with initial terms 1,3.A025229
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 3, with initial terms 1,3.
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 3, with initial terms 3,1.A025230
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 3, with initial terms 3,1.
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 3, with initial terms 2,3.A025231
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 3, with initial terms 2,3.
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 3, with initial terms 3,2.A025232
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 3, with initial terms 3,2.
Expansion of Product_{m>=1} (1 + q^m)^48.A025233
Expansion of Product_{m>=1} (1 + q^m)^48.
An L-tile is a 2 X 2 square with the upper 1 X 1 subsquare removed; no rotations are allowed. a(n) = number of tilings of a 4 X n rectangle using tiles that are either 1 X 1 squares or L-tiles.A025234
An L-tile is a 2 X 2 square with the upper 1 X 1 subsquare removed; no rotations are allowed. a(n) = number of tilings of a 4 X n rectangle using tiles that are either 1 X 1 squares or L-tiles.
a(n) = (1/2)*A014431(n+2).A025235
a(n) = (1/2)*A014431(n+2).
Duplicate of A014432.A025236
Duplicate of A014432.
Expansion of (1 -x -sqrt(1-2*x-11*x^2))/(6*x^2).A025237
Expansion of (1 -x -sqrt(1-2*x-11*x^2))/(6*x^2).
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-2)*a(2) for n >= 3.A025238
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-2)*a(2) for n >= 3.
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-2)*a(2) for n >= 3.A025239
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-2)*a(2) for n >= 3.
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-2)*a(2) for n >= 3.A025240
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-2)*a(2) for n >= 3.
Essentially same as A004148.A025241
Essentially same as A004148.
Generalized Catalan numbers A(x)^2 -(1+x)^2*A(x) +x*(2+x+x^2) =0.A025242
Generalized Catalan numbers A(x)^2 -(1+x)^2*A(x) +x*(2+x+x^2) =0.
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 1, 2, 1, 1.A025243
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 1, 2, 1, 1.
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 1, 1, 2, 2.A025244
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 1, 1, 2, 2.
a(n) = (1/2)*s(n+3), where s = A025244.A025245
a(n) = (1/2)*s(n+3), where s = A025244.
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 1, 0, 1, 1.A025246
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 1, 0, 1, 1.
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 2, 0, 1, 2.A025247
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 2, 0, 1, 2.
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 1, 0, 2, 2.A025248
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 1, 0, 2, 2.
a(n) = (1/2)*s(n+3), where s = A025248.A025249
a(n) = (1/2)*s(n+3), where s = A025248.
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 0, 1, 1, 0.A025250
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 0, 1, 1, 0.
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 0, 1, 2, 0.A025251
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 0, 1, 2, 0.
a(n) = (1/2)*s(n+3), where s = A025251.A025252
a(n) = (1/2)*s(n+3), where s = A025251.
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 0, 2, 1, 0.A025253
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 0, 2, 1, 0.
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 3, 1, 1, 3.A025254
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 3, 1, 1, 3.
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 1, 3, 1, 1.A025255
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 1, 3, 1, 1.
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 1, 1, 3, 3.A025256
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 1, 1, 3, 3.
(1/3)*s(n+3), where s = A025256.A025257
(1/3)*s(n+3), where s = A025256.
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 1, -1, 1, 1.A025258
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 1, -1, 1, 1.
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 2, -1, 1, 2.A025259
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 2, -1, 1, 2.
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 2, -2, 1, 2.A025260
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 2, -2, 1, 2.