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A210238 Triangle of multiplicities D(n) of multinomial coefficients corresponding to sequence A210237. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..63.

Sergei Viznyuk, C-program for the sequence.

EXAMPLE

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)

MATHEMATICA

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

CROSSREFS

Cf. A210237, A209936, A035206, A070289.

Sequence in context: A309220 A225632 A035206 * A209936 A213941 A181511

Adjacent sequences:  A210235 A210236 A210237 * A210239 A210240 A210241

KEYWORD

nonn,tabf

AUTHOR

Sergei Viznyuk, Mar 18 2012

STATUS

approved

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Last modified November 20 20:40 EST 2019. Contains 329347 sequences. (Running on oeis4.)