A145468 - OEIS

%I #7 Aug 15 2018 07:26:57

%S 1,0,2,2,3,4,7,8,13,16,23,28,40,48,66,82,107,132,171,208,266,324,406,

%T 494,614,740,912,1098,1338,1604,1945,2318,2793,3320,3972,4706,5605,

%U 6612,7840,9222,10882,12760,15004,17534,20542,23944,27949,32490,37813,43832

%N Convolution square of A003106.

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

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

%F 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.

%F 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

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

%t 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 *)

%o (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))}

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

%K nonn

%O 0,3

%A _Michael Somos_, Oct 11 2008