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A257490 Irregular triangle read by rows in which the n-th row lists multinomials (A036040) for partitions of 2n which have only even parts in Abramowitz-Stegun ordering.

%I A257490 #21 Aug 14 2019 16:09:03
%S A257490 1,1,3,1,15,15,1,28,35,210,105,1,45,210,630,1575,3150,945,1,66,495,
%T A257490 462,1485,13860,5775,13860,51975,51975,10395,1,91,1001,3003,3003,
%U A257490 45045,42042,105105,45045,630630,525525,315315,1576575,945945,135135
%N A257490 Irregular triangle read by rows in which the n-th row lists multinomials (A036040) for partitions of 2n which have only even parts in Abramowitz-Stegun ordering.
%C A257490 The length of row n is given by A000041(n).
%C A257490 Each entry in this irregular triangle is the quotient of the respective entries in A257468 and A096162, which is the multinomial called M_3 in Abramowitz-Stegun.
%C A257490 Has the same structure as the triangles in A036036, A096162, A115621 and A257468.
%H A257490 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], pp. 831-832.
%e A257490 Brackets group all partitions of the same length when there is more than one partition.
%e A257490 n/m  1    2          3           4    5
%e A257490 1:   1
%e A257490 2:   1    3
%e A257490 3:   1   15         15
%e A257490 4:   1  [28  35]   210         105
%e A257490 5:   1  [45 210]  [630 1575]  3150  945
%e A257490 ...
%e A257490 n = 6:  1 [66 495 462] [1485 13860 5775] [13860 51975] 51975  0395
%e A257490 Replacing the bracketed numbers by their sums yields the triangle of A156289.
%t A257490 (* triangle2574868[] and triangle096162[] are defined as functions triangle[] in the respective sequences A257468 and A096162 *)
%t A257490 triangle[n_] := triangle257468[n]/triangle096162[n]
%t A257490 a[n_] := Flatten[triangle[n]]
%t A257490 a[7] (* data *)
%Y A257490 Cf. A000041, A036040, A036036, A096162, A115621, A156289, A257468.
%K A257490 nonn,tabf
%O A257490 1,3
%A A257490 _Hartmut F. W. Hoft_, Apr 26 2015
%E A257490 Edited by _Wolfdieter Lang_, May 11 2015