A145468 Convolution square of A003106.

1, 0, 2, 2, 3, 4, 7, 8, 13, 16, 23, 28, 40, 48, 66, 82, 107, 132, 171, 208, 266, 324, 406, 494, 614, 740, 912, 1098, 1338, 1604, 1945, 2318, 2793, 3320, 3972, 4706, 5605, 6612, 7840, 9222, 10882, 12760, 15004, 17534, 20542, 23944, 27949, 32490, 37813, 43832

Offset

0,3

Links

Table of n, a(n) for n=0..49.

Formula

Expansion of H(x)^2 in powers of x where H() is a Rogers-Ramanujan function.

Euler transform of period 5 sequence [ 0, 2, 2, 0, 0, ...].

G.f.: (1 + Sum_{k>0} x^(k^2 - k) / ((1 - t) * (1 - t^2) * ... * (1 - t^k)))^2 = (Product_{k>0} (1 - x^(5*k - 2)) * (1 - x^(5*k -3)))^-2.

a(n) ~ (sqrt(5)-1) * exp(2*Pi*sqrt(2*n/15)) / (2^(7/4) * 3^(1/4) * 5^(3/4) * n^(3/4)) * (1 + (11*Pi/(15*sqrt(30)) - 3*sqrt(15/2)/(16*Pi)) / sqrt(n)). - Vaclav Kotesovec, Aug 14 2018

Example

q^11 + 2*q^71 + 2*q^101 + 3*q^131 + 4*q^161 + 7*q^191 + 8*q^221 + ...

Mathematica

nmax = 50; CoefficientList[Series[Product[1/((1 - x^(5*k - 2))*(1 - x^(5*k - 3)))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 14 2018 *)

Prog

(PARI) {a(n) = local(t); if( n<0, 0, t = 1 + x * O(x^n); polcoeff( sum(k= 1, (sqrt(4*n + 1) - 1) / 2, t *= x^(2*k) / (1 - x^k) * (1 + x * O(x^(n - k^2 - k))), 1)^2, n))}

Crossrefs

-A145466(5*n+2) = a(n).

Sequence in context: A215367 A276526 A091605 * A125554 A345867 A252796

Adjacent sequences: A145465 A145466 A145467 * A145469 A145470 A145471

Keyword

nonn

Author

Michael Somos, Oct 11 2008

Status

approved