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URL: https://oeis.org/A129358

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A129358
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)) ].
3
1, -5, -5, 70, -180, 770, -4760, 20840, -68085, 147890, -795, -1679855, 8378195, -25065005, 56439545, -145200415, 612604910, -2764023765, 10020060660, -28723695265, 67618167310, -128945409045, 137921330680, 375948665405, -3167538981120, 12823443150644, -38103903888575
OFFSET
0,2
COMMENTS
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).
FORMULA
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) ].
EXAMPLE
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)*...
Terms are divisible by 5 except at positions given by 25*A001318(n):
a(n) == 1 (mod 5) at n = [0, 125, 175, 550, 650,...,25*A036498(k),...];
a(n) == -1 (mod 5) at n = [25, 50, 300, 375, 875,...,25*A036499(k),...].
PROG
(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))}
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Apr 11 2007
STATUS
approved