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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.

%I A257468 #13 May 08 2018 15:11:56
%S A257468 1,1,6,1,15,90,1,28,70,420,2520,1,45,210,1260,3150,18900,113400,1,66,
%T A257468 495,924,2970,13860,34650,83160,207900,1247400,7484400,1,91,1001,3003,
%U A257468 6006,45045,84084,210210,270270,1261260,3153150,7567560,18918900,113513400,681080400
%N 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.
%C A257468 The row length sequence is A000041(n).
%C A257468 The triangle representation of this sequence has the same structure as the triangles in A036036 and A115621.
%C A257468 These multinomials, called M_1 by Abramowitz-Stegun on p. 831, are given in A036038.
%H A257468 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 A257468 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 A257468 n\m 1        2             3        4        5
%e A257468 1:  1
%e A257468 2:  1        6
%e A257468 3:  1       15            90
%e A257468 4:  1  [28  70]          420     2520
%e A257468 5:  1  [45 210]   [1260 3150]   18900   113400
%e A257468 ...
%e A257468 n = 6:  1 [66 495 924] [2970 13860 34650] [83160 207900] 1247400  7484400
%t A257468 (* row[] and triangle[] compute structured rows of the triangle *)
%t A257468 row[n_] := Map[Apply[Plus, #]!/Apply[Times, Map[Factorial, #]]&, GatherBy[2*IntegerPartitions[n], Length], {2}]
%t A257468 triangle[n_] := Map[row, Range[n]]
%t A257468 a[n_] := Flatten[triangle[n]]
%t A257468 a[7] (* data *)
%Y A257468 Cf. A000041, A036036, A036038, A115621.
%K A257468 nonn,tabf
%O A257468 1,3
%A A257468 _Hartmut F. W. Hoft_, Apr 25 2015
%E A257468 Edited. - _Wolfdieter Lang_, May 09 2015