A327870 -id:A327870 - cp's OEIS frontend

A327869 Sum T(n,k) of multinomials M(n; lambda), where lambda ranges over all partitions of n into distinct parts incorporating k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 1, 1, 1, 0, 1, 4, 3, 3, 1, 5, 4, 0, 4, 1, 16, 5, 10, 10, 5, 1, 82, 66, 75, 60, 15, 6, 1, 169, 112, 126, 35, 140, 21, 7, 1, 541, 456, 196, 336, 280, 224, 28, 8, 1, 2272, 765, 1548, 1848, 1386, 630, 336, 36, 9, 1, 17966, 15070, 15525, 16080, 14070, 3780, 1050, 480, 45, 10, 1

Offset: 0

Alois P. Heinz, Sep 28 2019

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

Number T(n,k) of set partitions of [n] with distinct block sizes and one of the block sizes is k. T(5,3) = 10: 123|45, 124|35, 125|34, 12|345, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234.

			Triangle T(n,k) begins:
      1;
      1,     1;
      1,     0,     1;
      4,     3,     3,     1;
      5,     4,     0,     4,     1;
     16,     5,    10,    10,     5,    1;
     82,    66,    75,    60,    15,    6,    1;
    169,   112,   126,    35,   140,   21,    7,   1;
    541,   456,   196,   336,   280,  224,   28,   8,  1;
   2272,   765,  1548,  1848,  1386,  630,  336,  36,  9,  1;
  17966, 15070, 15525, 16080, 14070, 3780, 1050, 480, 45, 10, 1;
  ...

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

Columns k=0-3 give: A007837, A327876, A327881, A328155.
Row sums give A327870.
T(2n,n) gives A328156.
Cf. A327801, A327884.

with(combinat):
T:= (n, k)-> add(multinomial(add(i, i=l), l[], 0),
             l=select(x-> nops(x)=nops({x[]}) and
             (k=0 or k in x), partition(n))):
seq(seq(T(n, k), k=0..n), n=0..11);
# second Maple program:
b:= proc(n, i, k) option remember; `if`(i*(i+1)/2 n!*(b(n$2, 0)-`if`(k=0, 0, b(n$2, k))):
seq(seq(T(n, k), k=0..n), n=0..11);

b[n_, i_, k_] := b[n, i, k] = If[i(i+1)/2 < n, 0, 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-1], k]/i!]]];
T[n_, k_] := n! (b[n, n, 0] - If[k == 0, 0, b[n, n, k]]);
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 28 2020, from 2nd Maple program *)

A320566 Expansion of e.g.f. exp(x) * Product_{k>=1} 1/(1 - x^k/k!).

Original entry on oeis.org

1, 2, 6, 23, 110, 617, 4035, 29927, 249926, 2316317, 23674841, 264329177, 3207278255, 42011308653, 591460307157, 8905905152798, 142897741683846, 2433947385964373, 43873382718719949, 834402502632550589, 16699964488044322205, 350869837371828862607, 7721899536993122262447

Offset: 0

Ilya Gutkovskiy, Oct 15 2018

nonn

Binomial transform of A005651.

Vaclav Kotesovec, Table of n, a(n) for n = 0..440
N. J. A. Sloane, Transforms

Cf. A005651, A320567, A327870.

Row sums of A327801.

seq(coeff(series(factorial(n)*exp(x)*mul((1-x^k/factorial(k))^(-1),k=1..n),x,n+1), x, n), n = 0 .. 22); # Muniru A Asiru, Oct 15 2018

nmax = 22; CoefficientList[Series[Exp[x] Product[1/(1 - x^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[x + Sum[Sum[x^(j k)/(k (j!)^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] Total[Apply[Multinomial, IntegerPartitions[k], {1}]], {k, 0, n}], {n, 0, 22}]

E.g.f.: exp(x + Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*(j!)^k)).
a(n) = Sum_{k=0..n} binomial(n,k)*A005651(k).
a(n) ~ exp(1) * A247551 * n!. - Vaclav Kotesovec, Jul 21 2019

cp's OEIS Frontend

A327869 Sum T(n,k) of multinomials M(n; lambda), where lambda ranges over all partitions of n into distinct parts incorporating k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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