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

Author

Sergei Viznyuk, Mar 18 2012

Keywords

• nonn
• tabf

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.

Examples

			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)
		

Links

• Sergei Viznyuk, C-program for the sequence.

Crossrefs

Cf. A210237, A209936, A035206, A070289.

Programs

Mathematica

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