cp's OEIS Frontend

A358334 Number of twice-partitions of n into odd-length partitions.

%I A358334 #12 Dec 30 2022 21:38:28
%S A358334 1,1,2,4,7,13,25,43,77,137,241,410,720,1209,2073,3498,5883,9768,16413,
%T A358334 26978,44741,73460,120462,196066,320389,518118,839325,1353283,2178764,
%U A358334 3490105,5597982,8922963,14228404,22609823,35875313,56756240,89761600,141410896,222675765
%N A358334 Number of twice-partitions of n into odd-length partitions.
%C A358334 A twice-partition of n (A063834) is a sequence of integer partitions, one of each part of an integer partition of n.
%H A358334 Andrew Howroyd, <a href="/A358334/b358334.txt">Table of n, a(n) for n = 0..1000</a>
%F A358334 G.f.: 1/Product_{k>=1} (1 - A027193(k)*x^k). - _Andrew Howroyd_, Dec 30 2022
%e A358334 The a(0) = 1 through a(5) = 13 twice-partitions:
%e A358334   ()  ((1))  ((2))     ((3))        ((4))           ((5))
%e A358334              ((1)(1))  ((111))      ((211))         ((221))
%e A358334                        ((2)(1))     ((2)(2))        ((311))
%e A358334                        ((1)(1)(1))  ((3)(1))        ((3)(2))
%e A358334                                     ((111)(1))      ((4)(1))
%e A358334                                     ((2)(1)(1))     ((11111))
%e A358334                                     ((1)(1)(1)(1))  ((111)(2))
%e A358334                                                     ((211)(1))
%e A358334                                                     ((2)(2)(1))
%e A358334                                                     ((3)(1)(1))
%e A358334                                                     ((111)(1)(1))
%e A358334                                                     ((2)(1)(1)(1))
%e A358334                                                     ((1)(1)(1)(1)(1))
%t A358334 twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
%t A358334 Table[Length[Select[twiptn[n],OddQ[Times@@Length/@#]&]],{n,0,10}]
%o A358334 (PARI)
%o A358334 P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
%o A358334 R(u,y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
%o A358334 seq(n) = {my(u=Vec(P(n,1)-P(n,-1))/2); Vec(R(u, 1), -(n+1))} \\ _Andrew Howroyd_, Dec 30 2022
%Y A358334 For multiset partitions of integer partitions: A356932, ranked by A356935.
%Y A358334 For odd length instead of lengths we have A358824.
%Y A358334 For odd sums instead of lengths we have A358825.
%Y A358334 For odd sums also we have A358827.
%Y A358334 For odd length also we have A358834.
%Y A358334 A000041 counts integer partitions.
%Y A358334 A027193 counts odd-length partitions, ranked by A026424.
%Y A358334 A055922 counts partitions with odd multiplicities, also odd parts A117958.
%Y A358334 A063834 counts twice-partitions, strict A296122, row-sums of A321449.
%Y A358334 Cf. A000219, A001970, A072233, A078408, A270995, A279374, A298118, A300300, A300301, A300647, A302243.
%K A358334 nonn
%O A358334 0,3
%A A358334 _Gus Wiseman_, Dec 01 2022
%E A358334 Terms a(21) and beyond from _Andrew Howroyd_, Dec 30 2022