A174068 Convolved with its aerated variant of two zeros between terms = A000041.

1, 1, 2, 2, 4, 5, 7, 9, 13, 17, 23, 29, 38, 48, 62, 77, 98, 121, 153, 187, 233, 283, 349, 422, 515, 620, 751, 900, 1083, 1291, 1544, 1832, 2180, 2576, 3050, 3590, 4234, 4965, 5830, 6813, 7971, 9286, 10824, 12572, 14608, 16921, 19600, 22640, 26150, 30130, 34709

Offset: 0

Author

Gary W. Adamson, Mar 06 2010

Keywords

• nonn

Comments

Considered k=3 in an infinite set of convolution sequences: (aerated with one zero, A174065; two zeros, A174068); such that A000041 =

(1, 1, 2, 3, 5, 7, 11,...) = (1, 1, 2, 2, 4, 5, 7, 9, 13, 17,...) *

(1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 4, 0, 0, 5, 0, 0, 7, 0, 0,...).

Examples

			The triangle heading and first few rows of the triangle =
1, 1, 2, 2, 4, 5, 7,...
1;
1;
2;
2, 1;
4, 1;
5, 2;
7, 2, 2;
...
G.f. = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 5*x^5 + 7*x^6 + 9*x^7 + 13*x^8 + 17*x^9 + ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Alois P. Heinz)

Crossrefs

Cf. A000041, A174065, A174066, A174067.

Programs

Maple

p:= combinat[numbpart]:
a:= proc(n) option remember; `if`(n=0, 1, p(n)-add(a(j)*
      `if`(irem(n-j, 3, 'r')>0, 0, a(r)), j=0..n-1))
    end:
seq(a(n), n=0..61);  # Alois P. Heinz, Jul 27 2019

Mathematica

a[n_] := a[n] = If[n == 0, 1, PartitionsP[n] - Sum[a[j]*If[Mod[n-j, 3] > 0, 0, a[(n-j)/3]], {j, 0, n-1}]];
a /@ Range[0, 61] (* Jean-François Alcover, May 17 2020, after Maple *)

Formula

Refer to A174065, and A174066, the case for k=3. The sequence = left border of a triangle generated from 3 rules: row sums = A000041; columns >1 are shifted down thrice from previous column; column terms are derived from self-convolution of left border, (with the left border placed at top as a heading).

A(x)*A(x^3) = A000041(x) for the generating functions. - R. J. Mathar, Mar 18 2010

Expansion of f(-x^3)/f(-x) * f(-x^27)/f(-x^9) * f(-x^243)/f(-x^27) * ... where f(-x) is a Ramanujan theta function. - Michael Somos, Jun 07 2012

a(n) ~ exp(Pi*sqrt(n/2)) / (2^(19/8) * 3^(1/8) * n^(7/8)). - Vaclav Kotesovec, Sep 24 2019

Extensions

More terms from R. J. Mathar, Mar 18 2010