A122928 Coefficients of a q-series inspired by Andrews and Ramanujan.

1, 1, 2, 3, 5, 8, 12, 18, 26, 37, 52, 72, 99, 134, 180, 240, 317, 416, 542, 702, 904, 1158, 1476, 1872, 2364, 2973, 3724, 4647, 5778, 7160, 8844, 10890, 13370, 16368, 19984, 24336, 29561, 35822, 43308, 52242, 62884, 75536, 90552, 108342, 129384, 154232

Offset: 0

Author

Michael Somos, Sep 19 2006

Keywords

• nonn

Links

• G. E. Andrews, Three aspects of partitions, Séminaire Lotharingien de Combinatoire, B25f (1990), 1 p.

Crossrefs

Cf. A098693(n)=a(n) if n>0.

Programs

Mathematica

nmax = 100; CoefficientList[Series[Product[(1-x^(12*k))*(1+x^(12*k-5))*(1+x^(12*k-7))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)

PARI

{a(n)=if(n<1, n==0, polcoeff( sum(k=1, sqrtint(n), x^k^2/(1+x^k)* prod(i=1, k, (1+x^i)^2/(1-x^(2*i-1))/(1-x^(2*i)), 1+x*O(x^(n-k^2)))), n))}

Formula

Euler transform of period 24 sequence [ 1, 1, 1, 1, 2, 1, 2, 1, 1, 0, 1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 1, 1, 1, 0, ...].

Given g.f. A(x), then B(x)=A(x)^2-A(x) satisfies 0=f(B(x), B(x^2)) where f(u, v)=(1+6*u)*v*(1+2*v)-u^2.

G.f.: {Sum_{k} q^(6k^2-k) }/{Sum_{k} (-1)^k q^((3k^2-k)/2) }.

G.f.: Product_{k>0} (1-q^(12k))(1+q^(12k-5))(1+q^(12k-7))/(1-q^k).

G.f.: 1+Sum_{k>0} Prod[i=1..k, (1+q^i)^2]*(1+q^k)*q^(k^2) /{(1-q)(1-q^2)...(1-q^(2k))}.

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(9/4) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Aug 31 2015