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A358831 Number of twice-partitions of n into partitions with weakly decreasing lengths.

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