A204383 G.f.: Product_{n>=1} (1 - A002203(n)*x^n + (-1)^n*x^(2*n))^3 where A002203(n) is the companion Pell numbers.

1, -6, -9, 70, 90, 0, -1411, -1722, 0, 490, 60534, 75222, 49, -21510, 0, -6067754, -7542180, 0, 2156110, 0, 81, 1420032740, 1764323886, 0, -504516870, -8118, 0, -50196874, -783087782910, -973096740630, -121, 278263575996, 0, 0, 27685627830, 0, 1024173639305948

Offset: 0

Author

Paul D. Hanna, Jan 14 2012

Keywords

• sign

Comments

a(A020757(n)) = 0 where A020757 lists numbers that are not the sum of two triangular numbers.

Examples

			G.f.: A(x) = 1 - 6*x - 9*x^2 + 70*x^3 + 90*x^4 - 1411*x^6 - 1722*x^7 +...
-log(A(x))/3 = 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)^3 * (1-6*x^2+x^4)^3 * (1-14*x^3-x^6)^3 * (1-34*x^4+x^8)^3 * (1-82*x^5-x^10)^3 * (1-198*x^6+x^12)^3 *...* (1 - A002203(n)*x^n + (-1)^n*x^(2*n))^3 *...
Positions of zeros form A020757:
[5,8,14,17,19,23,26,32,33,35,40,41,44,47,50,52,53,54,59,62,63,...].
		

Links

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

Crossrefs

Cf. A203861, A204382, A204384, A020757.

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, -3*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))^3, n)}

Formula

G.f.: exp( Sum_{n>=1} -3 * sigma(n) * A002203(n) * x^n/n ).