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A112500 Triangle of column sequences with a certain o.g.f. pattern.

%I A112500 #9 Aug 28 2019 17:29:44
%S A112500 1,1,1,1,4,1,1,11,10,1,1,26,60,20,1,1,57,282,225,35,1,1,120,1149,1882,
%T A112500 665,56,1,1,247,4272,13070,9107,1666,84,1,1,502,14932,79872,100751,
%U A112500 35028,3696,120,1,1,1013,49996,444902,957197,584325,113428,7470,165,1,1
%N A112500 Triangle of column sequences with a certain o.g.f. pattern.
%C A112500 The column o.g.f.s of this triangle appear as factors in the column o.g.f.s of triangle A008517 (second-order Eulerian numbers).
%H A112500 W. Lang, <a href="/A112500/a112500.txt">First ten rows.</a>
%F A112500 G.f. column k: G(k, x):= x^(k-1)/product((1-j*x)^(k-j+1), j=1..k), k>=1.
%F A112500 The column sequences start with A000012 (powers of 1), A000295 (Eulerian numbers), A112502-A112504.
%F A112500 a(n+k-1, k)=sum of product(binomial(n_j + k - 1, k - 1)*j^(n_j), j=1..k) with sum(n_j, j=1..k)=n, n_j >=0. There are binomial(n+k-1, k-1) terms of this sum and 1<=k<=n+1. a(n, k)=0 if n+1<k.
%e A112500 Rows: [1]; [1,1]; [1,4,1]; [1,11,10,1]; [1,26,60,20,1]; [1,57,282,225,35,1]; ...
%e A112500 a(4,3)= 60 = 6 + 12 + 9 + 12 + 9 + 12 from the binomial(4,2)=6 terms of the sum corresponding to (n_1,n_2,n_3) = (2,0,0), (0,2,0), (0,0,2), (1,1,0), (1,0,1) and (0,1,1).
%Y A112500 Cf. A112501 (row sums).
%K A112500 nonn,easy,tabl
%O A112500 0,5
%A A112500 _Wolfdieter Lang_, Oct 14 2005