A155826 Triangle T(n, k) = binomial(n, k) + binomial(k*(n-k), n) + 2*(-1)^n*StirlingS1(n, k)*StirlingS1(n, n-k), read by rows.

4, 1, 1, 1, 4, 1, 1, 15, 15, 1, 1, 76, 249, 76, 1, 1, 485, 3516, 3516, 485, 1, 1, 3606, 46623, 101354, 46623, 3606, 1, 1, 30247, 617541, 2388107, 2388107, 617541, 30247, 1, 1, 282248, 8416315, 51483931, 91651662, 51483931, 8416315, 282248, 1, 1, 2903049, 119667766, 1071669632, 3021085118, 3021085118, 1071669632, 119667766, 2903049, 1

Offset: 0

Author

Roger L. Bagula, Jan 28 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  4;
  1,      1;
  1,      4,       1;
  1,     15,      15,        1;
  1,     76,     249,       76,        1;
  1,    485,    3516,     3516,      485,        1;
  1,   3606,   46623,   101354,    46623,     3606,       1;
  1,  30247,  617541,  2388107,  2388107,   617541,   30247,     1;
  1, 282248, 8416315, 51483931, 91651662, 51483931, 8416315, 282248, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A048994, A342111.

Programs

Magma

A155826:= func< n,k | Binomial(n, k) + Binomial(k*(n-k), n) + 2*(-1)^n*StirlingFirst(n, k)*StirlingFirst(n, n-k) >;
[A155826(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 03 2021

Mathematica

T[n_, k_]:= Binomial[n, k] + Binomial[k*(n-k), n] + 2*(-1)^n*StirlingS1[n, k]*StirlingS1[n, n-k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 03 2021 *)

Sage

def A155826(n,k): return binomial(n, k) + binomial(k*(n-k), n) + 2*stirling_number1(n, k)*stirling_number1(n, n-k)
flatten([[A155826(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 03 2021

Formula

T(n, k) = binomial(n, k) + binomial(k*(n-k), n) + 2*(-1)^n*StirlingS1(n, k) * StirlingS1(n, n-k).

Sum_{k=0..n} T(n, k) = 2^n + 2*342111(n) + Sum_{k=0..n} binomial(k*(n-k), n). - G. C. Greubel, Jun 03 2021

Extensions

Edited by G. C. Greubel, Jun 03 2021