A147871 - OEIS

%I #22 Apr 14 2020 01:47:32

%S 1,1,1,3,4,7,10,15,24,37,49,73,105,142,208,294,391,538,752,988,1359,

%T 1812,2410,3232,4270,5598,7454,9721,12639,16625,21445,27649,35793,

%U 46235,59141,76215,96975,123262,157671,199625,252591,319792,403262,507682

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

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

%F 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

%e From _Petros Hadjicostas_, Apr 11 2020: (Start)

%e Let f(m) = A147665(m). Using the strict partitions of each n (see A000009), we get

%e a(1) = f(1) = 1,

%e a(2) = f(2) = 1,

%e a(3) = f(3) + f(1)*f(2) = 2 + 1*1 = 3,

%e a(4) = f(4) + f(1)*f(3) = 2 + 1*2 = 4,

%e a(5) = f(5) + f(1)*f(4) + f(2)*f(3) = 3 + 1*2 + 1*2 = 7,

%e 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,

%e 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)

%t (*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]]]];

%t P[x_, n_] := P[x, n] = Product[1 + f[m]*x^m, {m, 0, n}];

%t Take[CoefficientList[P[x, 45], x], 45] (* Program simplified by _Petros Hadjicostas_, Apr 13 2020 *)

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

%K nonn

%O 0,4

%A _Roger L. Bagula_, Nov 16 2008

%E Various sections edited by _Petros Hadjicostas_, Apr 11 2020