A266821 Expansion of Product_{k>=1} (1 + 3*x^k) / (1 - x^k).

1, 4, 8, 24, 44, 88, 176, 312, 544, 924, 1584, 2552, 4136, 6488, 10128, 15632, 23748, 35640, 53080, 78136, 114024, 165552, 237744, 339544, 481248, 678236, 949008, 1321840, 1830376, 2521688, 3456672, 4717208, 6406680, 8666448, 11672464, 15660528, 20934868

Offset: 0

Author

Vaclav Kotesovec, Jan 04 2016

Keywords

• nonn

Comments

Convolution of A000041 and A032308.

In general, for m > 0, if g.f. = Product_{k>=1} ((1 + m*x^k) / (1 - x^k)) then a(n) ~ sqrt(c) * exp(sqrt(2*c*n)) / (4*Pi*sqrt(m+1)*n), where c = 2*Pi^2/3 + log(m)^2 + 2*polylog(2, -1/m).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 5001 terms from Vaclav Kotesovec)

Crossrefs

Cf. A000041, A015128, A032308, A264686.

Column k=4 of A321884.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      (t-> b(t, min(t, i-1)))(n-i*j), j=1..n/i)*4 +b(n, i-1)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..44);  # Alois P. Heinz, Aug 28 2019

Mathematica

nmax = 40; CoefficientList[Series[Product[(1+3*x^k) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

PARI

{ my(n=40); Vec(prod(k=1, n, 4/(1-x^k) - 3 + O(x*x^n))) } \\ Andrew Howroyd, Dec 22 2017

Formula

a(n) ~ sqrt(c) * exp(sqrt(2*c*n)) / (8*Pi*n), where c = 2*Pi^2/3 + log(3)^2 + 2*polylog(2, -1/3) = 7.16861897522987077909937377164783326088308015803... .