%I
%S 1,1,6,1,15,90,1,28,70,420,2520,1,45,210,1260,3150,18900,113400,1,66,
%T 495,924,2970,13860,34650,83160,207900,1247400,7484400,1,91,1001,3003,
%U 6006,45045,84084,210210,270270,1261260,3153150,7567560,18918900,113513400,681080400
%N Triangle read by rows in which the nth row lists the multinomials A036038 for all partitions of 2n with only even parts in AbramowitzStegun ordering.
%C The row length sequence is A000041(n).
%C The triangle representation of this sequence has the same structure as the triangles in A036036 and A115621.
%C These multinomials, called M_1 by AbramowitzStegun on p. 831, are given in A036038.
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%e 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.
%e n\m 1 2 3 4 5
%e 1: 1
%e 2: 1 6
%e 3: 1 15 90
%e 4: 1 [28 70] 420 2520
%e 5: 1 [45 210] [1260 3150] 18900 113400
%e ...
%e n = 6: 1 [66 495 924] [2970 13860 34650] [83160 207900] 1247400 7484400
%t (* row[] and triangle[] compute structured rows of the triangle *)
%t row[n_] := Map[Apply[Plus, #]!/Apply[Times, Map[Factorial, #]]&, GatherBy[2*IntegerPartitions[n], Length], {2}]
%t triangle[n_] := Map[row, Range[n]]
%t a[n_] := Flatten[triangle[n]]
%t a[7] (* data *)
%Y Cf. A000041, A036036, A036038, A115621.
%K nonn,tabf
%O 1,3
%A _Hartmut F. W. Hoft_, Apr 25 2015
%E Edited.  _Wolfdieter Lang_, May 09 2015
