A257468 Triangle read by rows in which the n-th row lists the multinomials A036038 for all partitions of 2n with only even parts in Abramowitz-Stegun ordering.

1, 1, 6, 1, 15, 90, 1, 28, 70, 420, 2520, 1, 45, 210, 1260, 3150, 18900, 113400, 1, 66, 495, 924, 2970, 13860, 34650, 83160, 207900, 1247400, 7484400, 1, 91, 1001, 3003, 6006, 45045, 84084, 210210, 270270, 1261260, 3153150, 7567560, 18918900, 113513400, 681080400

Offset: 1

Author

Hartmut F. W. Hoft, Apr 25 2015

Keywords

• nonn
• tabf

Comments

The row length sequence is A000041(n).

The triangle representation of this sequence has the same structure as the triangles in A036036 and A115621.

These multinomials, called M_1 by Abramowitz-Stegun on p. 831, are given in A036038.

Examples

			The first six rows of the irregular triangle. The columns headings indicate the number of parts in the underlying partitions. Brackets group the multinomials for all partitions of the same length m when there is more than one partition.
n\m 1        2             3        4        5
1:  1
2:  1        6
3:  1       15            90
4:  1  [28  70]          420     2520
5:  1  [45 210]   [1260 3150]   18900   113400
...
n = 6:  1 [66 495 924] [2970 13860 34650] [83160 207900] 1247400  7484400
		

Links

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Crossrefs

Cf. A000041, A036036, A036038, A115621.

Programs

Mathematica

(* row[] and triangle[] compute structured rows of the triangle *)
row[n_] := Map[Apply[Plus, #]!/Apply[Times, Map[Factorial, #]]&, GatherBy[2*IntegerPartitions[n], Length], {2}]
triangle[n_] := Map[row, Range[n]]
a[n_] := Flatten[triangle[n]]
a[7] (* data *)

Extensions

Edited. - Wolfdieter Lang, May 09 2015