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

0,1

Links

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

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

Example

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;

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 *)

Prog

(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
(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

Crossrefs

Cf. A048994, A342111.

Sequence in context: A327939 A365837 A369669 * A273711 A340227 A351942

Adjacent sequences: A155823 A155824 A155825 * A155827 A155828 A155829

Keyword

nonn,tabl

Author

Roger L. Bagula, Jan 28 2009

Extensions

Edited by G. C. Greubel, Jun 03 2021

Status

approved