A204382 G.f.: Product_{n>=1} (1 - A002203(n)*x^n + (-1)^n*x^(2*n)) where A002203(n) is the companion Pell numbers.

1, -2, -7, -2, 1, 82, 34, 464, 198, -82, -1, 0, -39208, -16238, 6725, -551614, -228486, 95120, 0, 82, 6726, 0, 263673800, 109216786, -45239073, 0, 8957108166, 3706940654, -1536796802, -551614, -1, -109216786, 0, -18738638, -6726, -24954506565518, -10336495061766

Offset: 0

Author

Paul D. Hanna, Jan 14 2012

Keywords

• sign

Comments

a(A093519(n)) = 0 where A093519 lists numbers that are not equal to the sum of two generalized pentagonal numbers.

Examples

			G.f.: A(x) = 1 - 2*x - 7*x^2 - 2*x^3 + x^4 + 82*x^5 + 34*x^6 + 464*x^7 +...
-log(A(x)) = 1*2*x + 3*6*x^2/2 + 4*14*x^3/3 + 7*34*x^4/4 + 6*82*x^5/5 + 12*198*x^6/6 +...+ sigma(n)*A002203(n)*x^n/n +...
The g.f. equals the product:
A(x) = (1-2*x-x^2) * (1-6*x^2+x^4) * (1-14*x^3-x^6) * (1-34*x^4+x^8) * (1-82*x^5-x^10) * (1-198*x^6+x^12) *...* (1 - A002203(n)*x^n + (-1)^n*x^(2*n)) *...
Positions of zeros form A093519:
[11,18,21,25,32,39,43,46,49,54,60,65,67,68,74,76,81,87,88,90,...].
which are numbers that are not the sum of two generalized pentagonal numbers.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A203860, A204383, A204384, A203534, A093519, A002203.

Programs

PARI

/* Subroutine used in PARI programs below: */
{A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)), n)}

PARI

{a(n)=polcoeff(exp(sum(k=1, n, -sigma(k)*A002203(k)*x^k/k)+x*O(x^n)), n)}

PARI

{a(n)=polcoeff(prod(m=1, n, 1 - A002203(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n)), n)}

Formula

G.f.: exp( Sum_{n>=1} -sigma(n) * A002203(n) * x^n/n ).