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

0,5

LINKS

Alois P. Heinz, Rows n = 0..200, flattened

Wikipedia, Multinomial coefficients

Wikipedia, Partition (number theory)

FORMULA

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

EXAMPLE

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;

  ...

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);

CROSSREFS

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

Row sums give A005651.

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

Sequence in context: A265205 A263210 A298095 * A199667 A181163 A098867

Adjacent sequences:  A327800 A327801 A327802 * A327804 A327805 A327806

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Sep 25 2019

STATUS

approved

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Last modified February 19 19:18 EST 2020. Contains 332047 sequences. (Running on oeis4.)