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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. 7
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 (list; table; graph; refs; listen; history; text; internal format)
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);

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

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Last modified February 22 09:22 EST 2020. Contains 332133 sequences. (Running on oeis4.)