A185876 - OEIS

%I #16 Jul 20 2017 02:19:19

%S 1,5,6,15,29,21,35,85,99,56,70,195,285,259,126,126,385,645,735,574,

%T 252,210,686,1260,1645,1610,1134,462,330,1134,2226,3185,3570,3150,

%U 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 Fourth accumulation array of A051340, by antidiagonals.

%C A member of the accumulation chain A051340 < A141419 < A185874 < A185875 < A185876 < ... (See A144112 for the definition of accumulation array.)

%H G. C. Greubel, <a href="/A185876/b185876.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H 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 T(n,k) = (4*n+5*k+11)*C(k+2,3)*C(n+4,4)/20, k>=1, n>=1.

%e Northwest corner:

%e    1,   5,  15,   35,   70

%e    6,  29,  85,  195,  385

%e   21,  99, 285,  645, 1260

%e   56, 259, 735, 1645, 3185

%t f[n_,k_]:=k(1+k)n(1+n)(2+n)(5+4k+3n)/144;

%t TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185875 *)

%t Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

%t s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)

%t Factor[s[n,k]]  (* formula for A185876 *)

%t TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A185876 *)

%t Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

%Y Cf. A051340, A141419, A144112, A185874, A185875.

%Y Row 1: A000332, column 1: A000389.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Feb 05 2011