login
Twice partitioned numbers where the first partition is strict.
57

%I #28 Sep 18 2019 20:21:41

%S 1,1,2,5,8,18,34,65,109,223,386,698,1241,2180,3804,6788,11390,19572,

%T 34063,56826,96748,163511,272898,452155,755928,1244732,2054710,

%U 3382147,5534696,8992209,14733292,23763685,38430071,62139578,99735806,160183001,256682598

%N Twice partitioned numbers where the first partition is strict.

%C Number of sequences of integer partitions of the parts of some strict partition of n.

%C 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 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 G.f.: Product_{i>=1} (1 + A000041(i) * x^i).

%e 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 b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,

%p `if`(n=0, 1, b(n, i-1) +`if`(i>n, 0,

%p b(n-i, i-1)*combinat[numbpart](i))))

%p end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..50); # _Alois P. Heinz_, Apr 11 2016

%t With[{n = 50}, CoefficientList[Series[Product[(1 + PartitionsP[i] x^i), {i, 1, n}], {x, 0, n}], x]]

%Y Cf. A000009, A000041, A063834 (twice partitioned numbers), A270995, A279785, A327552, A327607.

%K nonn

%O 0,3

%A _Gus Wiseman_, Apr 10 2016