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A179313 Triangle T(n,k) read by rows: product of the compositorial weight of the k-th partition of n times A074664(.) applied to each part.

%I A179313 #4 Mar 31 2012 13:23:39
%S A179313 1,1,1,2,2,1,6,4,1,3,1,22,12,4,6,3,4,1,92,44,12,4,18,12,1,8,6,5,1,426,
%T A179313 184,44,24,66,36,12,6,24,24,4,10,10,6,1,2146,852,184,88,36,276,132,72,
%U A179313 18,12,88,72,24,24,1,30,40,10,12,15,7,1,11624,4292,852,368,264,1278,552
%N A179313 Triangle T(n,k) read by rows: product of the compositorial weight of the k-th partition of n times A074664(.) applied to each part.
%C A179313 Row n has A000041(n) entries. T(n,k) is the product of A074664(a_i) over all parts a_i
%C A179313 multiplied by the compositorial weight A048996(n,k) of the k-th partition (Abramowitz-Stegun order)
%C A179313 of n = sum_i a_i.
%C A179313 Summing also over the partitions with a common number of parts would create A127743.
%C A179313 In row n=4, for example, the partitions 3+1 and 2+2, each with 2 parts, are represented by
%C A179313 T(4,2)=4 and T(4,3)= 1 here, and the sum 4+1=5 of the entries is the single entry A127743(4,.).
%C A179313 In this sense, the table is a refinement of A127743.
%D A179313 M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, p. 831
%F A179313 T(n,k) = A048996(n,k) * A179380(n,k).
%F A179313 sum_{k=1..A000041(n)} T(n,k) = A000110(n).
%e A179313 T(6,3) represents the 3rd partition of 6, namely 2+4. A074664(2)*A074664(4) = 1*6 is multiplied
%e A179313 by the weight A048996([2,4]) = 2!/1!/1! =2, and T(6,3) =1*6*2=12.
%e A179313 T(6,5) represents the 5th partition of 6, namely 1+1+4. A074664(1)*A074664(1)*A074664(4) = 1*1*6 is multiplied
%e A179313 by the weight A048996([1,1,4]) = 3!/2!/1! =3, and T(6,5) =1*1*6*3.
%e A179313 T(7,6) represents the 6th partition of 7, namely 1+2+4. A074664(1)*A074664(2)*A074664(4) = 1*1*6 is multiplied
%e A179313 the weight A048996([1,2,4]) = 3!/1!/1!/1! =6, and T(7,6) =1*1*6*6.
%e A179313 The triangle starts
%e A179313 1;
%e A179313 1,1;
%e A179313 2,2,1;
%e A179313 6,4,1,3,1;
%e A179313 22,12,4,6,3,4,1;
%e A179313 92,44,12,4,18,12,1,8,6,5,1;
%e A179313 426,184,44,24,66,36,12,6,24,24,4,10,10,6,1;
%Y A179313 Cf. A000041, A127743
%K A179313 nonn,tabf
%O A179313 1,4
%A A179313 _Alford Arnold_, Jul 11 2010
%E A179313 Edited and extended by _R. J. Mathar_, Jul 16 2010