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Euler transform of partition triangle A008284.
18

%I #22 Feb 18 2023 16:01:55

%S 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 22,43,49,48,30,22,1,8,30,58,87,88,77,45,30,1,10,36,87,134,167,151,

%U 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 Euler transform of partition triangle A008284.

%C Number of multiset partitions of length-k integer partitions of n. - _Gus Wiseman_, Nov 09 2018

%H Alois P. Heinz, <a href="/A055884/b055884.txt">Rows n = 1..200, flattened</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%e From _Gus Wiseman_, Nov 09 2018: (Start)

%e Triangle begins:

%e 1

%e 1 2

%e 1 2 3

%e 1 4 4 5

%e 1 4 8 7 7

%e 1 6 12 16 12 11

%e 1 6 17 25 28 19 15

%e 1 8 22 43 49 48 30 22

%e 1 8 30 58 87 88 77 45 30

%e ...

%e The fifth row {1, 4, 8, 7, 7} counts the following multiset partitions:

%e {{5}} {{1,4}} {{1,1,3}} {{1,1,1,2}} {{1,1,1,1,1}}

%e {{2,3}} {{1,2,2}} {{1},{1,1,2}} {{1},{1,1,1,1}}

%e {{1},{4}} {{1},{1,3}} {{1,1},{1,2}} {{1,1},{1,1,1}}

%e {{2},{3}} {{1},{2,2}} {{2},{1,1,1}} {{1},{1},{1,1,1}}

%e {{2},{1,2}} {{1},{1},{1,2}} {{1},{1,1},{1,1}}

%e {{3},{1,1}} {{1},{2},{1,1}} {{1},{1},{1},{1,1}}

%e {{1},{1},{3}} {{1},{1},{1},{2}} {{1},{1},{1},{1},{1}}

%e {{1},{2},{2}}

%e (End)

%p h:= proc(n, i) option remember; expand(`if`(n=0, 1,

%p `if`(i<1, 0, h(n, i-1)+x*h(n-i, min(n-i, i)))))

%p end:

%p g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(

%p g(n, i-1, j-k)*x^(i*k)*binomial(coeff(h(n$2), x, i)+k-1, k), k=0..j))))

%p end:

%p b:= proc(n, i) option remember; expand(`if`(n=0, 1,

%p `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))

%p end:

%p T:= (n, k)-> coeff(b(n$2), x, k):

%p seq(seq(T(n,k), k=1..n), n=1..12); # _Alois P. Heinz_, Feb 17 2023

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

%t Table[Length[Join@@mps/@IntegerPartitions[n,{k}]],{n,5},{k,n}] (* _Gus Wiseman_, Nov 09 2018 *)

%Y Row sums give A001970.

%Y Main diagonal gives A000041.

%Y Columns k=1-2 give: A057427, A052928.

%Y T(n+2,n+1) gives A000070.

%Y T(2n,n) gives A360468.

%Y Cf. A055885, A055886, A360742.

%Y Cf. A000219, A007716, A008284, A255906, A317449, A317532, A317533, A320796, A320801, A320808.

%K nonn,tabl

%O 1,3

%A _Christian G. Bower_, Jun 09 2000