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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,...],...

%I A162506 #10 Jan 08 2016 12:51:32
%S A162506 1,1,3,6,12,23,42,77,132,236,390,664,1087,1782,2858,4601,7216,11344,
%T A162506 17650,27162,41632,63316,95717,143558,214644,318464,470879,691968,
%U A162506 1012866,1474434,2140606,3088874,4445440,6370142,9095564,12941289,18350398,25930984
%N 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,...],...
%C A162506 Equals row sums of triangle A162507.
%C A162506 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
%H A162506 Alois P. Heinz, <a href="/A162506/b162506.txt">Table of n, a(n) for n = 1..1000</a>
%F A162506 Convergent of an infinite product, a*b*c,...; a = [1,1,1,...], b =
%F A162506 [1,0,2,0,2,0,2,...], c = [1,0,0,3,0,0,3,0,0,3,...]; i.e. the infinite set of
%F A162506 sequences [1,...N,...,] interleaved with (N-2) adjacent zeros.
%F A162506 G.f.: x*Product(1+k*x^k/(1-x^k),k=1..infinity). - _Vladeta Jovovic_, Jul 24 2009
%e A162506 First few rows of the array =
%e A162506 1,...1,...1,...1,...1,...
%e A162506 1,...1,...3,...3,...5,...
%e A162506 1,...1,...3,...6,...8,...
%e A162506 1,...1,...3,...6,..12,...
%e A162506 1,...1,...3,...6,..12,...
%e A162506 ...tending to A162506: (1, 1, 3, 6, 12, 23, 42, 77, 132,...)
%p A162506 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A162506        b(n, i-1) +add(b(n-i*j, i-1)*i, j=1..n/i)))
%p A162506     end:
%p A162506 a:= n-> b(n-1, n-1):
%p A162506 seq(a(n), n=1..50);  # _Alois P. Heinz_, Feb 26 2013
%t A162506 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 *)
%Y A162506 Cf. A162507, A267007.
%K A162506 nonn
%O A162506 1,3
%A A162506 _Gary W. Adamson_, Jul 04 2009
%E A162506 More terms from _Vladeta Jovovic_, Jul 22 2009