A162506 Convergent of an infinite product, a*b*c,...; a = [1,1,1,...], b = [1,0,2,0,2,0,2,...], c = [1,0,0,3,0,0,3,0,0,3,...],...

1, 1, 3, 6, 12, 23, 42, 77, 132, 236, 390, 664, 1087, 1782, 2858, 4601, 7216, 11344, 17650, 27162, 41632, 63316, 95717, 143558, 214644, 318464, 470879, 691968, 1012866, 1474434, 2140606, 3088874, 4445440, 6370142, 9095564, 12941289, 18350398, 25930984

Offset

1,3

Comments

Equals row sums of triangle A162507.

With offset 0, sum of products of parts, counted without multiplicity, in all partitions of n. Sum of products of parts, counted with multiplicity, in all partitions of n is A006906. - Vladeta Jovovic, Jul 24 2009

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Formula

Convergent of an infinite product, a*b*c,...; a = [1,1,1,...], b =

[1,0,2,0,2,0,2,...], c = [1,0,0,3,0,0,3,0,0,3,...]; i.e. the infinite set of

sequences [1,...N,...,] interleaved with (N-2) adjacent zeros.

G.f.: x*Product(1+k*x^k/(1-x^k),k=1..infinity). - Vladeta Jovovic, Jul 24 2009

Example

First few rows of the array =
1,...1,...1,...1,...1,...
1,...1,...3,...3,...5,...
1,...1,...3,...6,...8,...
1,...1,...3,...6,..12,...
1,...1,...3,...6,..12,...
...tending to A162506: (1, 1, 3, 6, 12, 23, 42, 77, 132,...)

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1) +add(b(n-i*j, i-1)*i, j=1..n/i)))
    end:
a:= n-> b(n-1, n-1):
seq(a(n), n=1..50);  # Alois P. Heinz, Feb 26 2013

Mathematica

nmax = 50; Rest[CoefficientList[Series[x*Product[1+k*x^k/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jan 08 2016 *)

Crossrefs

Cf. A162507, A267007.

Sequence in context: A174201 A327546 A181844 * A328609 A227681 A055244

Adjacent sequences: A162503 A162504 A162505 * A162507 A162508 A162509

Keyword

nonn

Author

Gary W. Adamson, Jul 04 2009

Extensions

More terms from Vladeta Jovovic, Jul 22 2009

Status

approved