A147880 - OEIS

%I #37 Apr 14 2020 03:31:17

%S 1,1,1,3,5,8,12,21,30,50,75,110,169,249,361,539,757,1076,1583,2207,

%T 3121,4415,6184,8468,11775,16274,22314,30601,41745,56412,77008,103507,

%U 138383,186928,249855,333375,443898,588402,778276,1031126,1356945,1780645

%N Expansion of Product_{k > 0} (1 + A005229(k)*x^k).

%H Andrew Howroyd, <a href="/A147880/b147880.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: Product_{k > 0} (1 + A005229(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) = A005229(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.

%e From _Petros Hadjicostas_, Apr 10 2020: (Start)

%e Let f(m) = A005229(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) = 3 + 1*2 = 5,

%e a(5) = f(5) + f(1)*f(4) + f(2)*f(3) = 3 + 1*3 + 1*2 = 8,

%e a(6) = f(6) + f(1)*f(5) + f(2)*f(4) + f(1)*f(2)*f(3) = 4 + 1*3 + 1*3 + 1*1*2 = 12,

%e a(7) = f(7) + f(1)*f(6) + f(2)*f(5) + f(3)*f(4) + f(1)*f(2)*f(4) = 5 + 1*4 + 1*3 + 2*3 + 1*1*3 = 21. (End)

%t (*A005229*) f[n_Integer?Positive] := f[n] = f[ f[n - 2]] + f[n - f[n - 2]]; f[0] = 0; f[1] = f[2] = 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] (* Program simplified by _Petros Hadjicostas_, Apr 13 2020 *)

%o (PARI) \\ here B(n) is A005229 as vector.

%o B(n)={my(a=vector(n, i, 1)); for(n=3, n, a[n] = a[a[n-2]] + a[n-a[n-2]]); a}

%o seq(n)={my(v=B(n)); Vec(prod(k=1, n, 1 + v[k]*x^k + O(x*x^n)))} \\ _Andrew Howroyd_, Apr 10 2020

%Y Cf. A000009, A004001, A005229, A147559, A147654, A147655, A147665.

%K nonn

%O 0,4

%A _Roger L. Bagula_, Nov 16 2008

%E Various sections edited by _Joerg Arndt_ and _Petros Hadjicostas_, Apr 10 2020