A147869 - OEIS

%I #20 Apr 14 2020 20:04:47

%S 1,1,1,3,4,7,11,17,25,41,59,86,125,180,263,382,536,738,1073,1466,2028,

%T 2841,3889,5275,7211,9800,13249,17860,23948,31921,42864,56802,75115,

%U 99788,131239,172870,226789,296404,386745,504939,655227,849628,1101270

%N Expansion of Product_{k>0} (1 + A004001(k)*x^k).

%F a(n) = [x^n] Product_{k > 0} (1 + A004001(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) = A004001(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) = A004001(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) = 4 + 1*3 + 1*2 + 1*1*2 = 11,

%e a(7) = f(7) + f(1)*f(6) + f(2)*f(5) + f(3)*f(4) + f(1)*f(2)*f(4) = 4 + 1*4 + 1*3 + 2*2 + 1*1*2 = 17. (End)

%t f[0] = 0; f[1] = 1; f[2] = 1; f[n_] := f[n] = f[f[n - 1]] + f[n - f[n - 1]];

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

%t Take[CoefficientList[P[x, 45], x], 45]

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

%K nonn

%O 0,4

%A _Roger L. Bagula_, Nov 16 2008

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