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.

1, 1, 1, 3, 2, 1, 10, 9, 3, 1, 47, 40, 18, 4, 1, 246, 235, 100, 30, 5, 1, 1602, 1476, 705, 200, 45, 6, 1, 11481, 11214, 5166, 1645, 350, 63, 7, 1, 95503, 91848, 44856, 13776, 3290, 560, 84, 8, 1, 871030, 859527, 413316, 134568, 30996, 5922, 840, 108, 9, 1

Offset: 0

Author

Alois P. Heinz, Sep 25 2019

Keywords

• nonn
• tabl

Comments

Here we assume that every list of parts has at least one 0 because its addition does not change the value of the multinomial.

Examples

			Triangle T(n,k) begins:
      1;
      1,     1;
      3,     2,     1;
     10,     9,     3,     1;
     47,    40,    18,     4,    1;
    246,   235,   100,    30,    5,   1;
   1602,  1476,   705,   200,   45,   6,  1;
  11481, 11214,  5166,  1645,  350,  63,  7, 1;
  95503, 91848, 44856, 13776, 3290, 560, 84, 8, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• Wikipedia, Multinomial coefficients
• Wikipedia, Partition (number theory)

Crossrefs

Columns k=0-2 give: A005651, A327827, A327828.

Row sums give A320566.

T(2n,n) gives A266518.

T(n,n-1) gives A001477.

T(n+1,n-1) gives A045943.

Cf. A327869.

Programs

Maple

with(combinat):
T:= (n, k)-> add(multinomial(add(i, i=l), l[], 0), l=
             select(x-> k=0 or k in x, partition(n))):
seq(seq(T(n, k), k=0..n), n=0..10);
# second Maple program:
b:= proc(n, i, k) option remember; `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!))
    end:
T:= (n, k)-> n!*(b(n$2, 0)-`if`(k=0, 0, b(n$2, k))):
seq(seq(T(n, k), k=0..n), n=0..10);

Mathematica

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[n_, k_] := n! (b[n, n, 0] - If[k == 0, 0, b[n, n, k]]);
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 30 2020, from 2nd Maple program *)