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A134274 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(5)/M_3.

%I A134274 #17 Sep 25 2024 15:05:51
%S A134274 1,5,1,45,5,1,585,45,25,5,1,9945,585,225,45,25,5,1,208845,9945,2925,
%T A134274 2025,585,225,125,45,25,5,1,5221125,208845,49725,26325,9945,2925,2025,
%U A134274 1125,585,225,125,45,25,5,1,151412625,5221125,1044225,447525,342225
%N A134274 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(5)/M_3.
%C A134274 Partition number array M_3(5) = A134273 with each entry divided by the corresponding one of the partition number array M_3 = M_3(1) = A036040; in short M_3(5)/M_3.
%C A134274 The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
%C A134274 For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
%H A134274 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 A134274 Wolfdieter Lang, <a href="/A134274/a134274.txt">First 10 rows and more</a>.
%F A134274 a(n,k) = Product_{j=1..n} S2(5,j,1)^e(n,k,j) with S2(5,n,1) = A049029(n,1) = A007696(n) = (4*n-3)(!^4) (quadruple- or 4-factorials) 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 A134274 a(n,k) = A134273(n,k)/A036040(n,k) (division of partition arrays M_3(5) by M_3).
%e A134274 Triangle begins:
%e A134274   [1];
%e A134274   [5,1];
%e A134274   [45,5,1];
%e A134274   [585,45,25,5,1];
%e A134274   [9945,585,225,45,25,5,1];
%e A134274   ...
%Y A134274 Row sums A134276 (also of triangle A134275).
%Y A134274 Cf. A134150 (M_3(4)/M_3 array).
%K A134274 nonn,easy,tabf
%O A134274 1,2
%A A134274 _Wolfdieter Lang_, Nov 13 2007