A185876 Fourth accumulation array of A051340, by antidiagonals.

1, 5, 6, 15, 29, 21, 35, 85, 99, 56, 70, 195, 285, 259, 126, 126, 385, 645, 735, 574, 252, 210, 686, 1260, 1645, 1610, 1134, 462, 330, 1134, 2226, 3185, 3570, 3150, 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

Offset: 1

Author

Clark Kimberling, Feb 05 2011

Keywords

• nonn
• tabl

Comments

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

Examples

			Northwest corner:
   1,   5,  15,   35,   70
   6,  29,  85,  195,  385
  21,  99, 285,  645, 1260
  56, 259, 735, 1645, 3185
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
• Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 24.

Crossrefs

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

Row 1: A000332, column 1: A000389.

Programs

Mathematica

f[n_,k_]:=k(1+k)n(1+n)(2+n)(5+4k+3n)/144;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185875 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
Factor[s[n,k]]  (* formula for A185876 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A185876 *)
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

Formula

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