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%I #31 Apr 28 2020 07:33:06
%S 1,1,1,1,0,1,4,3,3,1,5,4,0,4,1,16,5,10,10,5,1,82,66,75,60,15,6,1,169,
%T 112,126,35,140,21,7,1,541,456,196,336,280,224,28,8,1,2272,765,1548,
%U 1848,1386,630,336,36,9,1,17966,15070,15525,16080,14070,3780,1050,480,45,10,1
%N Sum T(n,k) of multinomials M(n; lambda), where lambda ranges over all partitions of n into distinct parts incorporating k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C Here we assume that every list of parts has at least one 0 because its addition does not change the value of the multinomial.
%C Number T(n,k) of set partitions of [n] with distinct block sizes and one of the block sizes is k. T(5,3) = 10: 123|45, 124|35, 125|34, 12|345, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234.
%H Alois P. Heinz, <a href="/A327869/b327869.txt">Rows n = 0..140, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients">Multinomial coefficients</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e Triangle T(n,k) begins:
%e 1;
%e 1, 1;
%e 1, 0, 1;
%e 4, 3, 3, 1;
%e 5, 4, 0, 4, 1;
%e 16, 5, 10, 10, 5, 1;
%e 82, 66, 75, 60, 15, 6, 1;
%e 169, 112, 126, 35, 140, 21, 7, 1;
%e 541, 456, 196, 336, 280, 224, 28, 8, 1;
%e 2272, 765, 1548, 1848, 1386, 630, 336, 36, 9, 1;
%e 17966, 15070, 15525, 16080, 14070, 3780, 1050, 480, 45, 10, 1;
%e ...
%p with(combinat):
%p T:= (n, k)-> add(multinomial(add(i, i=l), l[], 0),
%p l=select(x-> nops(x)=nops({x[]}) and
%p (k=0 or k in x), partition(n))):
%p seq(seq(T(n, k), k=0..n), n=0..11);
%p # second Maple program:
%p b:= proc(n, i, k) option remember; `if`(i*(i+1)/2<n, 0,
%p `if`(n=0, 1, `if`(i<2, 0, b(n, i-1, `if`(i=k, 0, k)))+
%p `if`(i=k, 0, b(n-i, min(n-i, i-1), k)/i!)))
%p end:
%p T:= (n, k)-> n!*(b(n$2, 0)-`if`(k=0, 0, b(n$2, k))):
%p seq(seq(T(n, k), k=0..n), n=0..11);
%t b[n_, i_, k_] := b[n, i, k] = If[i(i+1)/2 < n, 0, If[n==0, 1, If[i<2, 0, b[n, i-1, If[i==k, 0, k]]] + If[i==k, 0, b[n-i, Min[n-i, i-1], k]/i!]]];
%t T[n_, k_] := n! (b[n, n, 0] - If[k == 0, 0, b[n, n, k]]);
%t Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 28 2020, from 2nd Maple program *)
%Y Columns k=0-3 give: A007837, A327876, A327881, A328155.
%Y Row sums give A327870.
%Y T(2n,n) gives A328156.
%Y Cf. A327801, A327884.
%K nonn,tabl
%O 0,7
%A _Alois P. Heinz_, Sep 28 2019
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