A129356 G.f.: A(x) = Product_{n>=1} [ (1-x)^3*(1 + 3x + 6x^2 +...+ n(n+1)/2*x^(n-1)) ].

1, -3, -3, 15, -15, 66, -261, 618, -1155, 1040, 2361, -11616, 23733, -27027, 29394, -132318, 545790, -1383459, 2418896, -3383679, 4278462, -3127320, -8332866, 42021990, -99069516, 160683318, -200247795, 214883010, -345461022, 1184850729, -3966311448, 9899287254, -18787986009

Offset

0,2

Comments

a(k) != 0 (mod 3) at k = 9*A001318(n) for n>=0, where A001318 are the generalized pentagonal numbers: m(3m-1)/2, m=0,+-1,+-2,...; a(k) == 1 (mod 3) at k = 9*A036498(n) (n>=0); a(k) == -1 (mod 3) at k = 9*A036499(n) (n>=0).

Links

Table of n, a(n) for n=0..32.

Formula

G.f.: A(x) = Product_{n>=1} [ 1 - (n+1)(n+2)/2*x^n + n(n+2)*x^(n+1) - n(n+1)/2*x^(n+2) ].

Example

A(x) = (1-3x+3x^2-x^3)(1-6x^2+8x^3-3x^4)(1-10x^3+15x^4-6x^5)*...
*( 1 - (n+1)(n+2)/2*x^n + n(n+2)*x^(n+1) - n(n+1)/2*x^(n+2) )*...
Terms are divisible by 3 except at positions given by:
a(n) == 1 (mod 3) at n = [0, 45, 63, 198, 234, 459,...,9*A036498(k),..];
a(n) == -1 (mod 3) at n = [9, 18, 108, 135, 315, 360,..,9*A036499(k),..].

Prog

(PARI) {a(n)=if(n==0, 1, polcoeff(prod(k=1, n, (1-x)^3*sum(j=1, k, j*(j+1)/2*x^(j-1)) +x*O(x^n)), n))}

Crossrefs

Cf. A129355, A129357, A129358; A001318, A036498, A036499.

Sequence in context: A086116 A100735 A362531 * A290344 A217858 A185275

Adjacent sequences: A129353 A129354 A129355 * A129357 A129358 A129359

Keyword

sign

Author

Paul D. Hanna, Apr 10 2007

Status

approved