A174065 Convolved with its aerated variant = A000041.

1, 1, 1, 2, 3, 4, 5, 7, 9, 12, 15, 19, 25, 31, 38, 48, 60, 73, 89, 109, 133, 161, 193, 232, 279, 333, 395, 470, 558, 658, 775, 912, 1071, 1254, 1464, 1708, 1991, 2313, 2681, 3107, 3595, 4149, 4782, 5506, 6331, 7268, 8330, 9538, 10912, 12462, 14213, 16199

Offset: 0

Author

Gary W. Adamson, Mar 06 2010

Keywords

• nonn

Comments

A000041 = (1, 1, 2, 3, 5, 7, ...) = (1, 1, 1, 2, 3, 4, 5, 7, ...) * (1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 7, 0, 9, 0, ...).

The sequence diverges from A100853 after 16 terms; and is a conjectured Euler transform of A035263: (1, 0, 1, 1, 1, 0, 1, 0, 1, ...).

Examples

			Heading at top, with triangle A174066 underneath (the generator for A174065):
1, 1, 1, 2, 3, 4,.... = heading
1;................... = 1
1;................... = 1
1, 1;................ = 2
2, 1;................ = 3
3, 1, 1;............. = 5
4, 2, 1;............. = 7
5, 3, 1, 2;.......... = 11
7, 4, 2, 2;.......... = 15
9, 5, 3, 2, 3;....... = 22
...
... where terms in the left column are the result of the two rules: multiply heading * left column, and row sums = partition numbers.
Thus leftmost term in column 8 must be 7 = 15 - (4 + 2 + 2). Then the 7 is placed in its spot in the left column and as the next heading term.
		

Links

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

Crossrefs

Cf. A000041, A327726, A373219, A373220, A373221.

Cf. A000009, A115362, 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, 2, 'r')=1, 0, a(r)), j=0..n-1))
    end:
seq(a(n), n=0..61);  # Alois P. Heinz, Jul 27 2019

Mathematica

nmax = 60; CoefficientList[Series[Product[QPochhammer[-1, x^(4^j)]/2, {j, 0, Log[nmax]/Log[4]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 24 2019 *)

Formula

Aerate and convolve sequences are generated by triangles (in this case A174066) in which ongoing terms are placed in the left column and at the top as a heading. Columns >1 are shifted down k times (k=2) in this case corresponding to (k-1) interpolated zeros. Next term in left column = n-th term in the "target sequence" S(n) (in this case A000041) minus (sum of terms in n-th row for columns >1). Place the latter term in the heading filling in missing terms.

G.f.: Product_{i>=1, j>=0} (1 + x^(i*4^j)). - Ilya Gutkovskiy, Sep 23 2019

a(n) ~ exp(2*Pi*sqrt(n)/3) / (2^(11/8) * 3^(3/4) * n^(7/8)). - Vaclav Kotesovec, Sep 24 2019

From Seiichi Manyama, May 31 2024: (Start)

G.f.: Product_{k>=1} (1 + x^k)^(valuation(k,4) + 1).

Let A(x) be the g.f. of this sequence, and B(x) be the g.f. of A000009, then B(x) = A(x)/A(x^4). (End)

Extensions

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

Offset corrected by Alois P. Heinz, Jul 27 2019