A174068 - OEIS

%I #26 May 17 2020 07:23:59

%S 1,1,2,2,4,5,7,9,13,17,23,29,38,48,62,77,98,121,153,187,233,283,349,

%T 422,515,620,751,900,1083,1291,1544,1832,2180,2576,3050,3590,4234,

%U 4965,5830,6813,7971,9286,10824,12572,14608,16921,19600,22640,26150,30130,34709

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

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

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

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

%H Vaclav Kotesovec, <a href="/A174068/b174068.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..5000 from Alois P. Heinz)

%F 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).

%F A(x)*A(x^3) = A000041(x) for the generating functions. - _R. J. Mathar_, Mar 18 2010

%F 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

%F a(n) ~ exp(Pi*sqrt(n/2)) / (2^(19/8) * 3^(1/8) * n^(7/8)). - _Vaclav Kotesovec_, Sep 24 2019

%e The triangle heading and first few rows of the triangle =

%e 1, 1, 2, 2, 4, 5, 7,...

%e 1;

%e 1;

%e 2;

%e 2, 1;

%e 4, 1;

%e 5, 2;

%e 7, 2, 2;

%e ...

%e 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 + ...

%p p:= combinat[numbpart]:

%p a:= proc(n) option remember; `if`(n=0, 1, p(n)-add(a(j)*

%p       `if`(irem(n-j, 3, 'r')>0, 0, a(r)), j=0..n-1))

%p     end:

%p seq(a(n), n=0..61);  # _Alois P. Heinz_, Jul 27 2019

%t 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}]];

%t a /@ Range[0, 61] (* _Jean-François Alcover_, May 17 2020, after Maple *)

%Y Cf. A000041, A174065, A174066, A174067.

%K nonn

%O 0,3

%A _Gary W. Adamson_, Mar 06 2010

%E More terms from _R. J. Mathar_, Mar 18 2010