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%I #3 Mar 30 2012 18:37:03
%S 1,-5,-5,70,-180,770,-4760,20840,-68085,147890,-795,-1679855,8378195,
%T -25065005,56439545,-145200415,612604910,-2764023765,10020060660,
%U -28723695265,67618167310,-128945409045,137921330680,375948665405,-3167538981120,12823443150644,-38103903888575
%N G.f.: A(x) = Product_{n>=1} [ (1-x)^5*(1 + 5x + 15x^2 +...+ n(n+1)(n+2)(n+3)/4!*x^(n-1)) ].
%C a(k) != 0 (mod 5) at k = 25*A001318(n) for n>=0, where A001318 are the generalized pentagonal numbers: m(3m-1)/2, m=0,+-1,+-2,...; a(k) == 1 (mod 5) at k = 25*A036498(n) (n>=0); a(k) == -1 (mod 5) at k = 25*A036499(n) (n>=0).
%F G.f.: A(x) = Product_{n>=1} [ 1 - (n+1)(n+2)(n+3)(n+4)/4!*x^n + 4n(n+2)(n+3)(n+4)/4!*x^(n+1) - 6n(n+1)(n+3)(n+4)/4!*x^(n+2) + 4n(n+1)(n+2)(n+4)/4!*x^(n+3) - n(n+1)(n+2)(n+3)/4!*x^(n+4) ].
%e A(x) = (1-5x+10x^2-10x^3+5x^4-x^5)*(1-15x^2+40x^3-45x^4+24x^5-5x^6)*(1-35x^3+105x^4-126x^5+70x^6-15x^7)*(1-70x^4+224x^5-280x^6+160x^7-35x^8)*...
%e Terms are divisible by 5 except at positions given by 25*A001318(n):
%e a(n) == 1 (mod 5) at n = [0, 125, 175, 550, 650,...,25*A036498(k),...];
%e a(n) == -1 (mod 5) at n = [25, 50, 300, 375, 875,...,25*A036499(k),...].
%o (PARI) {a(n)=if(n==0,1,polcoeff(prod(k=1,n,(1-x)^5*sum(j=1,k,binomial(j+3,4)*x^(j-1)) +x*O(x^n)),n))}
%Y Cf. A129355, A129356, A129357; A001318, A036498, A036499.
%K sign
%O 0,2
%A _Paul D. Hanna_, Apr 11 2007
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