A174148 Triangle read by rows: T(n,k) = binomial(n,k)*(binomial(n-1,k-1)*binomial(n+1,k+1) + binomial(n-1,k)*binomial(n+1,k)), with T(0,0) = 1.

1, 1, 1, 1, 12, 1, 1, 42, 42, 1, 1, 100, 360, 100, 1, 1, 195, 1700, 1700, 195, 1, 1, 336, 5775, 14000, 5775, 336, 1, 1, 532, 15876, 75950, 75950, 15876, 532, 1, 1, 792, 37632, 312816, 617400, 312816, 37632, 792, 1, 1, 1125, 79920, 1058400, 3630312, 3630312, 1058400, 79920, 1125, 1

Offset

0,5

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Formula

T(n,k) = (Product_{i=-1..1} binomial(n+i,k+i)) + (Product_{i=-1..1} binomial(n+i,n-k+i)) for n > 0.

T(n,k) = T(n,n-k).

Example

Triangle read by rows:
  1;
  1,   1;
  1,  12,     1;
  1,  42,    42,      1;
  1, 100,   360,    100,      1;
  1, 195,  1700,   1700,    195,      1;
  1, 336,  5775,  14000,   5775,    336,     1;
  1, 532, 15876,  75950,  75950,  15876,   532,   1;
  1, 792, 37632, 312816, 617400, 312816, 37632, 792, 1;
  ...

Mathematica

t[n_, m_, q_] = If[n == 0 || n == 1, 1, Product[Binomial[n + i, m + i], {i, - Floor[q/2], Floor[q/2]}] + Product[Binomial[n + i, n - m + i], { i, -Floor[q/2], Floor[q/2]}]];
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 0, 10, 2}]

Prog

(PARI) T(n, k) = if(n==0, 1, binomial(n, k)*(binomial(n-1, k-1)*binomial(n+1, k+1) + binomial(n-1, k)*binomial(n+1, k))) \\ Andrew Howroyd, Nov 19 2025

Crossrefs

Column 1 is A007586.

Row sums are 2*A277188(n-1).

Sequence in context: A186432 A176489 A174039 * A155495 A157273 A350729

Adjacent sequences: A174145 A174146 A174147 * A174149 A174150 A174151

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 09 2010

Extensions

Edited by Andrew Howroyd, Nov 19 2025

Status

approved