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A134150 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(4)/M_3.

%I A134150 #16 Sep 25 2024 15:06:10
%S A134150 1,4,1,28,4,1,280,28,16,4,1,3640,280,112,28,16,4,1,58240,3640,1120,
%T A134150 784,280,112,64,28,16,4,1,1106560,58240,14560,7840,3640,1120,784,448,
%U A134150 280,112,64,28,16,4,1,24344320,1106560,232960,101920,78400,58240,14560,7840
%N A134150 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(4)/M_3.
%C A134150 The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
%C A134150 For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
%C A134150 Partition number array M_3(4) = A134149 with each entry divided by the corresponding one of the partition number array M_3 = M_3(1) = A036040; in short, M_3(4)/M_3.
%H A134150 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].
%H A134150 Wolfdieter Lang, <a href="/A134150/a134150.txt">First 10 rows and more</a>.
%F A134150 a(n,k) = Product_{j=1..n} S2(4,j,1)^e(n,k,j) with S2(4,n,1) = A035469(n,1) = A007559(n) = (3*n-2)!!! and with the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n.
%F A134150 a(n,k) = A134149(n,k)/A036040(n,k) (division of partition arrays M_3(4) by M_3).
%e A134150 Triangle begins:
%e A134150   [1];
%e A134150   [4,1];
%e A134150   [28,4,1];
%e A134150   [280,28,16,4,1];
%e A134150   [3640,280,112,28,16,4,1];
%e A134150   ...
%e A134150 a(4,3)=16 from the third (k=3) partition (2^2) of 4: (4)^2 = 16, because S2(4,2,1) = 4!! = 4*1 = 4.
%Y A134150 Cf. A134145 (M_3(3)/M_3 array).
%Y A134150 Cf. A134152 (row sums, also of triangle A134151).
%K A134150 nonn,easy,tabf
%O A134150 1,2
%A A134150 _Wolfdieter Lang_, Nov 13 2007