A210237 -id:A210237 - cp's OEIS frontend

A210238 Triangle of multiplicities D(n) of multinomial coefficients corresponding to sequence A210237.

1, 2, 1, 3, 6, 1, 4, 12, 6, 12, 1, 5, 20, 20, 30, 30, 20, 1, 6, 30, 30, 15, 60, 120, 20, 60, 90, 30, 1, 7, 42, 42, 42, 105, 210, 105, 245, 420, 140, 105, 210, 42, 1, 8, 56, 56, 224, 28, 336, 336, 280, 168, 168, 840, 420, 1120, 70, 1120, 560, 168, 420, 56, 1

Offset: 1

Sergei Viznyuk, Mar 18 2012

Multiplicity D(n) of multinomial coefficient M(n) is the number of ways the same value of M(n)=n!/(m1!*m2!*..*mk!) is obtained by distributing n identical balls into k distinguishable bins.
Differs from A209936 after a(21).
Differs from A035206 after a(36).
The checksum relationship: sum(M(n)*D(n)) = k^n
The number of terms per row (for each value of n starting with n=1) forms sequence A070289.

			1
2, 1
3, 6, 1
4, 12, 6, 12, 1
5, 20, 20, 30, 30, 20, 1
6, 30, 30, 15, 60, 120, 20, 60, 90, 30, 1
7, 42, 42, 42, 105, 210, 105, 245, 420, 140, 105, 210, 42, 1
Thus for n=3 (third row) the same value of multinomial coefficient follows from the following combinations:
3!/(3!0!0!) 3!/(0!3!0!) 3!/(0!0!3!) (i.e. multiplicity=3)
3!/(2!1!0!) 3!/(2!0!1!) 3!/(0!2!1!) 3!/(0!1!2!) 3!/(1!0!2!) 3!/(1!2!0!)  (i.e. multiplicity=6)
3!/(1!1!1!) (i.e. multiplicity=1)

Sergei Viznyuk, C-program for the sequence.

Cf. A210237, A209936, A035206, A070289.

Table[Last/@Tally[Multinomial@@@Compositions[k,k]],{k,8}] (* Wouter Meeussen, Mar 09 2013 *)

A309972 Product of multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n.

Original entry on oeis.org

1, 1, 2, 18, 6912, 216000000, 1632586752000000000, 498266101635303733401600000000000, 1140494258799407218656986754465090350453096448000000000000000

Offset: 0

Alois P. Heinz, Aug 25 2019

nonn

			a(3) = M(3;3) * M(3;2,1) * M(3;1,1,1) = 1 * 3 * 6 = 18.

Alois P. Heinz, Table of n, a(n) for n = 0..14
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)

Rightmost terms in rows of A309951.

Cf. A000041, A005651, A036038, A078760, A210237.

b:= proc(n, i) option remember; `if`(n=0 or i=1, [n!], [map(t->
      binomial(n, i)*t, b(n-i, min(n-i, i)))[], b(n, i-1)[]])
    end:
a:= n-> mul(i, i=b(n$2)):
seq(a(n), n=0..9);  # Alois P. Heinz, Aug 25 2019

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {n!}, Join[Binomial[n, i] #& /@ b[n - i, Min[n - i, i]], b[n, i - 1]]];
a[n_] := Times @@ b[n, n];
a /@ Range[0, 9] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

a(n) = Product_{k=1..A000041(n)} A036038(n,k).

a(n) = A309951(n,A000041(n)).

cp's OEIS Frontend

A210238 Triangle of multiplicities D(n) of multinomial coefficients corresponding to sequence A210237.

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