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A185876 Fourth accumulation array of A051340, by antidiagonals.

%I A185876 #16 Jul 20 2017 02:19:19
%S A185876 1,5,6,15,29,21,35,85,99,56,70,195,285,259,126,126,385,645,735,574,
%T A185876 252,210,686,1260,1645,1610,1134,462,330,1134,2226,3185,3570,3150,
%U A185876 2058,792,495,1770,3654,5586,6860,6930,5670,3498,1287,715,2640,5670,9114,11956,13230,12390,9570,5643,2002,1001,3795,8415,14070
%N A185876 Fourth accumulation array of A051340, by antidiagonals.
%C A185876 A member of the accumulation chain A051340 < A141419 < A185874 < A185875 < A185876 < ... (See A144112 for the definition of accumulation array.)
%H A185876 G. C. Greubel, <a href="/A185876/b185876.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A185876 Johann Cigler, <a href="https://arxiv.org/abs/1611.05252">Some elementary observations on Narayana polynomials and related topics</a>, arXiv:1611.05252 [math.CO], 2016. See p. 24.
%F A185876 T(n,k) = (4*n+5*k+11)*C(k+2,3)*C(n+4,4)/20, k>=1, n>=1.
%e A185876 Northwest corner:
%e A185876    1,   5,  15,   35,   70
%e A185876    6,  29,  85,  195,  385
%e A185876   21,  99, 285,  645, 1260
%e A185876   56, 259, 735, 1645, 3185
%t A185876 f[n_,k_]:=k(1+k)n(1+n)(2+n)(5+4k+3n)/144;
%t A185876 TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185875 *)
%t A185876 Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%t A185876 s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
%t A185876 Factor[s[n,k]]  (* formula for A185876 *)
%t A185876 TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A185876 *)
%t A185876 Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%Y A185876 Cf. A051340, A141419, A144112, A185874, A185875.
%Y A185876 Row 1: A000332, column 1: A000389.
%K A185876 nonn,tabl
%O A185876 1,2
%A A185876 _Clark Kimberling_, Feb 05 2011