A203860 G.f.: Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) where Lucas(n) = A000204(n).

1, -1, -4, -1, 1, 11, 7, 25, 18, -11, -1, 0, -325, -199, 122, -1364, -843, 550, 0, 11, 123, 0, 39650, 24476, -15126, 0, 271443, 164194, -103682, -1364, -1, -24476, 0, -9349, -123, -20633239, -12752043, 7881225, -843, 0, -226965629, -141422125, 88114450, 0, 1

Offset

0,3

Comments

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

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Formula

G.f.: exp( Sum_{n>=1} -sigma(n) * A000204(n) * x^n/n ).

Example

G.f.: A(x) = 1 - x - 4*x^2 - x^3 + x^4 + 11*x^5 + 7*x^6 + 25*x^7 +...
-log(A(x)) = x + 3*3*x^2/2 + 4*4*x^3/3 + 7*7*x^4/4 + 6*11*x^5/5 + 12*18*x^6/6 +...+ sigma(n)*A000204(n)*x^n/n +...
The g.f. equals the product:
A(x) = (1-x-x^2) * (1-3*x^2+x^4) * (1-4*x^3-x^6) * (1-7*x^4+x^8) * (1-11*x^5-x^10) * (1-18*x^6+x^12) *...* (1 - Lucas(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.

Prog

(PARI) /* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
(PARI) {a(n)=polcoeff(exp(sum(k=1, n, -sigma(k)*Lucas(k)*x^k/k)+x*O(x^n)), n)}
(PARI) {a(n)=polcoeff(prod(m=1, n, 1 - Lucas(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n)), n)}

Crossrefs

Cf. A203861, A203850, A156234, A093519.

Sequence in context: A145271 A232774 A331969 * A147564 A090981 A087903

Adjacent sequences: A203857 A203858 A203859 * A203861 A203862 A203863

Keyword

sign

Author

Paul D. Hanna, Jan 07 2012

Status

approved