A327803 Sum T(n,k) of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts that form a set of size k; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

1, 0, 1, 0, 3, 0, 7, 3, 0, 31, 16, 0, 121, 125, 0, 831, 711, 60, 0, 5041, 5915, 525, 0, 42911, 46264, 6328, 0, 364561, 438681, 67788, 0, 3742453, 4371085, 753420, 12600, 0, 39916801, 49321745, 8924685, 166320, 0, 486891175, 588219523, 113501784, 2966040

Offset: 0

Author

Alois P. Heinz, Sep 25 2019

Keywords

• nonn
• tabf

Examples

			Triangle T(n,k) begins:
  1;
  0,       1;
  0,       3;
  0,       7,       3;
  0,      31,      16;
  0,     121,     125;
  0,     831,     711,     60;
  0,    5041,    5915,    525;
  0,   42911,   46264,   6328;
  0,  364561,  438681,  67788;
  0, 3742453, 4371085, 753420, 12600;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened
• Wikipedia, Multinomial coefficients
• Wikipedia, Partition (number theory)

Crossrefs

Columns k=0-2 give: A000007, A061095, A327826.

Row sums give A005651.

Cf. A000217, A003056, A022915, A131632 (when the parts are distinct), A226874.

Programs

Maple

with(combinat):
T:= (n, k)-> add(multinomial(add(i, i=l), l[], 0), l=
             select(x-> nops({x[]})=k, partition(n))):
seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..14);
# second Maple program:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(x^signum(j)*b(n-i*j, i-1)*
      combinat[multinomial](n, n-i*j, i$j), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..14);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i<1, 0, Sum[x^Sign[j]*b[n - i*j, i-1]*multinomial[n, Join[{n-i*j}, Table[i, {j}]]], {j, 0, n/i}]]]];
T[n_] := CoefficientList[b[n, n], x];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, May 06 2020, after 2nd Maple program *)

Formula

T(n*(n+1)/2,n) = T(A000217(n),n) = A022915(n).