A217480 G.f.: [ Sum_{n>=0} (-1)^n * x^(n*(3*n+2)) * (1 + x^(2*n+1)) ]^6.

1, 6, 15, 20, 15, 0, -29, -60, -66, -60, -51, 0, 60, 84, 135, 160, 81, 60, 70, -60, -120, -108, -255, -240, -50, -126, -120, 100, 30, 60, 331, 180, 90, 360, 195, 0, 120, -180, -306, 80, -72, -312, -120, -300, -360, 60, -51, -240, 171, 300, 150, 260, 135, -60, 240, 420

Offset

0,2

Comments

The number of zero terms is infinite.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..10000

Formula

G.f. A(x) satisfies:

A(x^2)^(1/6) = Sum_{n>=0} x^n / Product_{k=0..n} (1 + x^(2*k+1)), by an identity due to Ramanujan.

Example

G.f.: A(x) = 1 + 6*x + 15*x^2 + 20*x^3 + 15*x^4 - 29*x^6 - 60*x^7 - 66*x^8 +...
By definition, A(x) = B(x)^6 where
B(x) = 1 + x - x^5 - x^8 + x^16 + x^21 - x^33 - x^40 + x^56 + x^65 - x^85 - x^96 + x^120 + x^133 - x^161 - x^176 + x^208 + x^225 +...
By Ramanujan's identity,
B(x^2) = 1/(1+x) + x/((1+x)*(1+x^3)) + x^2/((1+x)*(1+x^3)*(1+x^5)) + x^3/((1+x)*(1+x^3)*(1+x^5)*(1+x^7)) + x^4/((1+x)*(1+x^3)*(1+x^5)*(1+x^7)*(1+x^9)) +...
Zero-valued coefficients in g.f. A(x) = B(x)^6 are found at positions:
[5, 11, 35, 69, 75, 151, 177, 180, 226, 516, 539, 728, 922, 2544, 3320, 3936, 4796, 5658, 5702, 6968, 7887, 8603, 9472, ...];
higher powers of B(x) have at most a finite number of zero coefficients.

Prog

(PARI) {a(n)=local(A=sum(m=0, n, (-1)^m*x^(m*(3*m+2))*(1+x^(2*m+1))+x*O(x^n))); polcoeff(A^6, n)}
for(n=0, 60, print1(a(n), ", "))

Crossrefs

Sequence in context: A105139 A002599 A001484 * A335417 A173680 A230209

Adjacent sequences: A217477 A217478 A217479 * A217481 A217482 A217483

Keyword

sign

Author

Paul D. Hanna, Oct 20 2012

Status

approved