A156581 Triangle T(n, k, m) = (m+2)^(k*(n-k)) with m = 15, read by rows.

1, 1, 1, 1, 17, 1, 1, 289, 289, 1, 1, 4913, 83521, 4913, 1, 1, 83521, 24137569, 24137569, 83521, 1, 1, 1419857, 6975757441, 118587876497, 6975757441, 1419857, 1, 1, 24137569, 2015993900449, 582622237229761, 582622237229761, 2015993900449, 24137569, 1

Offset

0,5

Links

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

Formula

T(n, k, m) = b(n, m)/(b(k, m)*b(n-k, m)) with b(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} binomial(j-1, i)*(k+1)^i ), b(n, 0) = n!, and m = 15.

T(n, k, m) = (m+2)^(k*(n-k)) with m = 15. - G. C. Greubel, Jun 28 2021

Example

Triangle begins as:
  1;
  1,       1;
  1,      17,          1;
  1,     289,        289,            1;
  1,    4913,      83521,         4913,          1;
  1,   83521,   24137569,     24137569,      83521,       1;
  1, 1419857, 6975757441, 118587876497, 6975757441, 1419857, 1;

Mathematica

(* First program *)
b[n_, k_]:= b[n, k]= If[k==0, n!, Product[Sum[Binomial[j-1, i]*(k+1)^i, {i, 0, j-1}], {j, n}]];
T[n_, k_, m_]:= T[n, k, m]= b[n, m]/(b[k, m]*b[n-k, m]);
Table[T[n, k, 15], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jun 28 2021 *)
(* Second program *)
T[n_, k_, m_]:= (m+2)^(k*(n-k)); Table[T[n, k, 15], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 28 2021 *)

Prog

(Magma)
A156581:= func< n, k, m | (m+2)^(k*(n-k)) >;
[A156581(n, k, 15): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 28 2021
(Sage)
def A156581(n, k, m): return (m+2)^(k*(n-k))
flatten([[A156581(n, k, 15) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 28 2021

Crossrefs

Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), this sequence (m=15).

Sequence in context: A176794 A176244 A022180 * A015143 A172196 A056110

Adjacent sequences: A156578 A156579 A156580 * A156582 A156583 A156584

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 10 2009

Extensions

Edited by G. C. Greubel, Jun 28 2021

Status

approved