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

%I

%S 1,-3,-3,15,-15,66,-261,618,-1155,1040,2361,-11616,23733,-27027,29394,

%T -132318,545790,-1383459,2418896,-3383679,4278462,-3127320,-8332866,

%U 42021990,-99069516,160683318,-200247795,214883010,-345461022,1184850729,-3966311448,9899287254,-18787986009

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

%C 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).

%F 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) ].

%e A(x) = (1-3x+3x^2-x^3)(1-6x^2+8x^3-3x^4)(1-10x^3+15x^4-6x^5)*...

%e *( 1 - (n+1)(n+2)/2*x^n + n(n+2)*x^(n+1) - n(n+1)/2*x^(n+2) )*...

%e Terms are divisible by 3 except at positions given by:

%e a(n) == 1 (mod 3) at n = [0, 45, 63, 198, 234, 459,...,9*A036498(k),..];

%e a(n) == -1 (mod 3) at n = [9, 18, 108, 135, 315, 360,..,9*A036499(k),..].

%o (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))}

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

%K sign

%O 0,2

%A _Paul D. Hanna_, Apr 10 2007

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Last modified May 11 08:56 EDT 2021. Contains 343784 sequences. (Running on oeis4.)