%I #9 Dec 31 2022 20:30:09
%S 1,1,3,6,14,26,56,102,205,372,708,1260,2345,4100,7388,12819,22603,
%T 38658,67108,113465,193876,324980,547640,909044,1516609,2495023,
%U 4118211,6726997,11002924,17836022,28948687,46604803,75074397,120134298,192188760,305709858,486140940
%N Number of twice-partitions of n into partitions with weakly decreasing lengths.
%C A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.
%H Andrew Howroyd, <a href="/A358831/b358831.txt">Table of n, a(n) for n = 0..500</a>
%e The a(1) = 1 through a(4) = 14 twice-partitions:
%e (1) (2) (3) (4)
%e (11) (21) (22)
%e (1)(1) (111) (31)
%e (2)(1) (211)
%e (11)(1) (1111)
%e (1)(1)(1) (2)(2)
%e (3)(1)
%e (11)(2)
%e (21)(1)
%e (11)(11)
%e (111)(1)
%e (2)(1)(1)
%e (11)(1)(1)
%e (1)(1)(1)(1)
%t twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
%t Table[Length[Select[twiptn[n],GreaterEqual@@Length/@#&]],{n,0,10}]
%o (PARI)
%o P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
%o seq(n) = {my(g=Vec(P(n,y)-1), v=[1]); for(k=1, n, my(p=g[k], u=v); v=vector(k+1); v[1] = 1 + O(x*x^n); for(j=1, k, v[1+j] = (v[j] + if(j<k, u[1+j] - u[j]))/(1 - polcoef(p,j)*x^k))); Vec(v[1+n])} \\ _Andrew Howroyd_, Dec 31 2022
%Y This is the semi-ordered case of A141199.
%Y For constant instead of weakly decreasing lengths we have A306319.
%Y For distinct instead of weakly decreasing lengths we have A358830.
%Y A063834 counts twice-partitions, strict A296122, row-sums of A321449.
%Y A196545 counts p-trees, enriched A289501.
%Y Cf. A000041, A000219, A001970, A061260, A072233, A271619, A279787, A358836.
%K nonn
%O 0,3
%A _Gus Wiseman_, Dec 03 2022
%E Terms a(26) and beyond from _Andrew Howroyd_, Dec 31 2022