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A271619 Twice partitioned numbers where the first partition is strict.

%I A271619 #28 Sep 18 2019 20:21:41
%S A271619 1,1,2,5,8,18,34,65,109,223,386,698,1241,2180,3804,6788,11390,19572,
%T A271619 34063,56826,96748,163511,272898,452155,755928,1244732,2054710,
%U A271619 3382147,5534696,8992209,14733292,23763685,38430071,62139578,99735806,160183001,256682598
%N A271619 Twice partitioned numbers where the first partition is strict.
%C A271619 Number of sequences of integer partitions of the parts of some strict partition of n.
%C A271619 This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1, g(n) = -A000041(n). - _Seiichi Manyama_, Nov 15 2018
%H A271619 Seiichi Manyama, <a href="/A271619/b271619.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..5000 from Alois P. Heinz)
%F A271619 G.f.: Product_{i>=1} (1 + A000041(i) * x^i).
%e A271619 a(6)=34: {(6);(5)(1),(51);(4)(2),(42);(4)(11),(41)(1),(411);(33);(3)(2)(1),(31)(2),(32)(1),(321);(3)(11)(1),(31)(11),(311)(1),(3111);(22)(2),(222);(21)(2)(1),(22)(11),(211)(2),(221)(1),(2211);(21)(11)(1),(111)(2)(1),(211)(11),(1111)(2),(2111)(1),(21111);(111)(11)(1),(1111)(11),(11111)(1),(111111)}
%p A271619 b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,
%p A271619       `if`(n=0, 1, b(n, i-1) +`if`(i>n, 0,
%p A271619        b(n-i, i-1)*combinat[numbpart](i))))
%p A271619     end:
%p A271619 a:= n-> b(n$2):
%p A271619 seq(a(n), n=0..50);  # _Alois P. Heinz_, Apr 11 2016
%t A271619 With[{n = 50}, CoefficientList[Series[Product[(1 + PartitionsP[i] x^i), {i, 1, n}], {x, 0, n}], x]]
%Y A271619 Cf. A000009, A000041, A063834 (twice partitioned numbers), A270995, A279785, A327552, A327607.
%K A271619 nonn
%O A271619 0,3
%A A271619 _Gus Wiseman_, Apr 10 2016