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A327801 Sum T(n,k) of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts incorporating k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A327801 #33 Apr 30 2020 07:19:04
%S A327801 1,1,1,3,2,1,10,9,3,1,47,40,18,4,1,246,235,100,30,5,1,1602,1476,705,
%T A327801 200,45,6,1,11481,11214,5166,1645,350,63,7,1,95503,91848,44856,13776,
%U A327801 3290,560,84,8,1,871030,859527,413316,134568,30996,5922,840,108,9,1
%N A327801 Sum T(n,k) of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts incorporating k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C A327801 Here we assume that every list of parts has at least one 0 because its addition does not change the value of the multinomial.
%H A327801 Alois P. Heinz, <a href="/A327801/b327801.txt">Rows n = 0..140, flattened</a>
%H A327801 Wikipedia, <a href="https://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients">Multinomial coefficients</a>
%H A327801 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%e A327801 Triangle T(n,k) begins:
%e A327801       1;
%e A327801       1,     1;
%e A327801       3,     2,     1;
%e A327801      10,     9,     3,     1;
%e A327801      47,    40,    18,     4,    1;
%e A327801     246,   235,   100,    30,    5,   1;
%e A327801    1602,  1476,   705,   200,   45,   6,  1;
%e A327801   11481, 11214,  5166,  1645,  350,  63,  7, 1;
%e A327801   95503, 91848, 44856, 13776, 3290, 560, 84, 8, 1;
%e A327801   ...
%p A327801 with(combinat):
%p A327801 T:= (n, k)-> add(multinomial(add(i, i=l), l[], 0), l=
%p A327801              select(x-> k=0 or k in x, partition(n))):
%p A327801 seq(seq(T(n, k), k=0..n), n=0..10);
%p A327801 # second Maple program:
%p A327801 b:= proc(n, i, k) option remember; `if`(n=0, 1,
%p A327801       `if`(i<2, 0, b(n, i-1, `if`(i=k, 0, k)))+
%p A327801       `if`(i=k, 0, b(n-i, min(n-i, i), k)/i!))
%p A327801     end:
%p A327801 T:= (n, k)-> n!*(b(n$2, 0)-`if`(k=0, 0, b(n$2, k))):
%p A327801 seq(seq(T(n, k), k=0..n), n=0..10);
%t A327801 b[n_, i_, k_] := b[n, i, k] = 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], k]/i!]];
%t A327801 T[n_, k_] := n! (b[n, n, 0] - If[k == 0, 0, b[n, n, k]]);
%t A327801 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 30 2020, from 2nd Maple program *)
%Y A327801 Columns k=0-2 give: A005651, A327827, A327828.
%Y A327801 Row sums give A320566.
%Y A327801 T(2n,n) gives A266518.
%Y A327801 T(n,n-1) gives A001477.
%Y A327801 T(n+1,n-1) gives A045943.
%Y A327801 Cf. A327869.
%K A327801 nonn,tabl
%O A327801 0,4
%A A327801 _Alois P. Heinz_, Sep 25 2019