A147871 Expansion of Product_{k > 0} (1 + A147665(k)*x^k).

1, 1, 1, 3, 4, 7, 10, 15, 24, 37, 49, 73, 105, 142, 208, 294, 391, 538, 752, 988, 1359, 1812, 2410, 3232, 4270, 5598, 7454, 9721, 12639, 16625, 21445, 27649, 35793, 46235, 59141, 76215, 96975, 123262, 157671, 199625, 252591, 319792, 403262, 507682

Offset: 0

Author

Roger L. Bagula, Nov 16 2008

Keywords

• nonn

Examples

			From _Petros Hadjicostas_, Apr 11 2020: (Start)
Let f(m) = A147665(m). Using the strict partitions of each n (see A000009), we get
a(1) = f(1) = 1,
a(2) = f(2) = 1,
a(3) = f(3) + f(1)*f(2) = 2 + 1*1 = 3,
a(4) = f(4) + f(1)*f(3) = 2 + 1*2 = 4,
a(5) = f(5) + f(1)*f(4) + f(2)*f(3) = 3 + 1*2 + 1*2 = 7,
a(6) = f(6) + f(1)*f(5) + f(2)*f(4) + f(1)*f(2)*f(3) = 3 + 1*3 + 1*2 + 1*1*2 = 10,
a(7) = f(7) + f(1)*f(6) + f(2)*f(5) + f(3)*f(4) + f(1)*f(2)*f(4) = 3 + 1*3 + 1*3 + 2*2 + 1*1*2 = 15. (End)
		

Crossrefs

Cf. A000009, A004001, A147559, A147654, A147655, A147665, A147880.

Programs

Mathematica

(*A147665*) f[0] = 0; f[1] = 1; f[2] = 1; f[n_] := f[n] = f[f[n - 1]] + If[Mod[ n, 3] == 0, f[f[n/3]], If[Mod[n, 3] == 1, f[f[(n - 1)/3]], f[n - f[(n - 2)/3]]]];
P[x_, n_] := P[x, n] = Product[1 + f[m]*x^m, {m, 0, n}];
Take[CoefficientList[P[x, 45], x], 45] (* Program simplified by Petros Hadjicostas, Apr 13 2020 *)

Formula

a(n) = [x^n] Product_{k > 0} (1 + A147665(k)*x^k).

a(n) = Sum_{(b_1,...,b_n)} f(1)^b_1 * f(2)^b_2 * ... * f(n)^b_n, where f(m) = A147665(m), and the sum is taken over all lists (b_1,...,b_n) with b_j in {0,1} and Sum_{j=1..n} j*b_j = n. - Petros Hadjicostas, Apr 11 2020

Extensions

Various sections edited by Petros Hadjicostas, Apr 11 2020