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A055884 Euler transform of partition triangle A008284.

%I A055884 #22 Feb 18 2023 16:01:55
%S A055884 1,1,2,1,2,3,1,4,4,5,1,4,8,7,7,1,6,12,16,12,11,1,6,17,25,28,19,15,1,8,
%T A055884 22,43,49,48,30,22,1,8,30,58,87,88,77,45,30,1,10,36,87,134,167,151,
%U A055884 122,67,42,1,10,45,113,207,270,296,247,185,97,56,1,12,54,155,295,448,510,507,394,278,139,77
%N A055884 Euler transform of partition triangle A008284.
%C A055884 Number of multiset partitions of length-k integer partitions of n. - _Gus Wiseman_, Nov 09 2018
%H A055884 Alois P. Heinz, <a href="/A055884/b055884.txt">Rows n = 1..200, flattened</a>
%H A055884 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%e A055884 From _Gus Wiseman_, Nov 09 2018: (Start)
%e A055884 Triangle begins:
%e A055884    1
%e A055884    1   2
%e A055884    1   2   3
%e A055884    1   4   4   5
%e A055884    1   4   8   7   7
%e A055884    1   6  12  16  12  11
%e A055884    1   6  17  25  28  19  15
%e A055884    1   8  22  43  49  48  30  22
%e A055884    1   8  30  58  87  88  77  45  30
%e A055884    ...
%e A055884 The fifth row {1, 4, 8, 7, 7} counts the following multiset partitions:
%e A055884   {{5}}   {{1,4}}     {{1,1,3}}       {{1,1,1,2}}         {{1,1,1,1,1}}
%e A055884           {{2,3}}     {{1,2,2}}      {{1},{1,1,2}}       {{1},{1,1,1,1}}
%e A055884          {{1},{4}}   {{1},{1,3}}     {{1,1},{1,2}}       {{1,1},{1,1,1}}
%e A055884          {{2},{3}}   {{1},{2,2}}     {{2},{1,1,1}}      {{1},{1},{1,1,1}}
%e A055884                      {{2},{1,2}}    {{1},{1},{1,2}}     {{1},{1,1},{1,1}}
%e A055884                      {{3},{1,1}}    {{1},{2},{1,1}}    {{1},{1},{1},{1,1}}
%e A055884                     {{1},{1},{3}}  {{1},{1},{1},{2}}  {{1},{1},{1},{1},{1}}
%e A055884                     {{1},{2},{2}}
%e A055884 (End)
%p A055884 h:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p A055884       `if`(i<1, 0, h(n, i-1)+x*h(n-i, min(n-i, i)))))
%p A055884     end:
%p A055884 g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
%p A055884       g(n, i-1, j-k)*x^(i*k)*binomial(coeff(h(n$2), x, i)+k-1, k), k=0..j))))
%p A055884     end:
%p A055884 b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p A055884      `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
%p A055884     end:
%p A055884 T:= (n, k)-> coeff(b(n$2), x, k):
%p A055884 seq(seq(T(n,k), k=1..n), n=1..12);  # _Alois P. Heinz_, Feb 17 2023
%t A055884 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A055884 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A055884 Table[Length[Join@@mps/@IntegerPartitions[n,{k}]],{n,5},{k,n}] (* _Gus Wiseman_, Nov 09 2018 *)
%Y A055884 Row sums give A001970.
%Y A055884 Main diagonal gives A000041.
%Y A055884 Columns k=1-2 give: A057427, A052928.
%Y A055884 T(n+2,n+1) gives A000070.
%Y A055884 T(2n,n) gives A360468.
%Y A055884 Cf. A055885, A055886, A360742.
%Y A055884 Cf. A000219, A007716, A008284, A255906, A317449, A317532, A317533, A320796, A320801, A320808.
%K A055884 nonn,tabl
%O A055884 1,3
%A A055884 _Christian G. Bower_, Jun 09 2000