A147880 Expansion of Product_{k > 0} (1 + A005229(k)*x^k).

1, 1, 1, 3, 5, 8, 12, 21, 30, 50, 75, 110, 169, 249, 361, 539, 757, 1076, 1583, 2207, 3121, 4415, 6184, 8468, 11775, 16274, 22314, 30601, 41745, 56412, 77008, 103507, 138383, 186928, 249855, 333375, 443898, 588402, 778276, 1031126, 1356945, 1780645

Offset

0,4

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Formula

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

Example

From Petros Hadjicostas, Apr 10 2020: (Start)
Let f(m) = A005229(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) = 3 + 1*2 = 5,
a(5) = f(5) + f(1)*f(4) + f(2)*f(3) = 3 + 1*3 + 1*2 = 8,
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,
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)

Mathematica

(*A005229*) f[n_Integer?Positive] := f[n] = f[ f[n - 2]] + f[n - f[n - 2]]; f[0] = 0; f[1] = f[2] = 1;
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 *)

Prog

(PARI) \\ here B(n) is A005229 as vector.
B(n)={my(a=vector(n, i, 1)); for(n=3, n, a[n] = a[a[n-2]] + a[n-a[n-2]]); a}
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

Crossrefs

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

Sequence in context: A055606 A295832 A147879 * A276527 A020643 A131354

Adjacent sequences: A147877 A147878 A147879 * A147881 A147882 A147883

Keyword

nonn

Author

Roger L. Bagula, Nov 16 2008

Extensions

Various sections edited by Joerg Arndt and Petros Hadjicostas, Apr 10 2020

Status

approved