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

1,2

Comments

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

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.

Formula

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

Example

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

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

Crossrefs

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

Row 1: A000332, column 1: A000389.

Sequence in context: A115908 A247962 A241307 * A356496 A091020 A019070

Adjacent sequences: A185873 A185874 A185875 * A185877 A185878 A185879

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 05 2011

Status

approved