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

0,4

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.

Links

Alois P. Heinz, Rows n = 0..140, flattened

Wikipedia, Multinomial coefficients

Wikipedia, Partition (number theory)

Example

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;
  ...

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 *)

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.

Sequence in context: A187105 A116071 A214622 * A320578 A267836 A319669

Adjacent sequences: A327798 A327799 A327800 * A327802 A327803 A327804

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 25 2019

Status

approved