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

Author

Michael Somos, Oct 11 2008

Keywords

• nonn

Examples

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

Crossrefs

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

Programs

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 *)

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))}

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