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A320733 Number of partitions of n with two sorts of part 1 which are introduced in ascending order.

%I A320733 #12 Dec 07 2020 09:06:03
%S A320733 1,1,3,6,13,26,54,108,219,439,882,1766,3539,7081,14172,28351,56716,
%T A320733 113443,226908,453833,907698,1815424,3630893,7261829,14523725,
%U A320733 29047513,58095121,116190338,232380810,464761759,929523710,1859047619,3718095507,7436191301
%N A320733 Number of partitions of n with two sorts of part 1 which are introduced in ascending order.
%H A320733 Alois P. Heinz, <a href="/A320733/b320733.txt">Table of n, a(n) for n = 0..3322</a>
%F A320733 G.f.: ((1 - x)/(1 - 2*x)) * Product_{k>=2} 1/(1 - x^k). - _Ilya Gutkovskiy_, Dec 03 2019
%p A320733 b:= proc(n, i) option remember; `if`(n=0 or i<2, add(
%p A320733       Stirling2(n, j), j=0..2), add(b(n-i*j, i-1), j=0..n/i))
%p A320733     end:
%p A320733 a:= n-> b(n$2):
%p A320733 seq(a(n), n=0..40);
%t A320733 b[n_, i_] := b[n, i] = If[n == 0 || i < 2, Sum[StirlingS2[n, j], {j, 0, 2}], Sum[b[n - i j, i - 1], {j, 0, n/i}]];
%t A320733 a[n_] := b[n, n];
%t A320733 a /@ Range[0, 40] (* _Jean-François Alcover_, Dec 07 2020, after _Alois P. Heinz_ *)
%Y A320733 Column k=2 of A292745.
%Y A320733 Cf. A002865, A011782, A090764.
%K A320733 nonn
%O A320733 0,3
%A A320733 _Alois P. Heinz_, Oct 20 2018