A363398 Triangle read by rows. T(n, k) = [x^k] P(n, x), where P(n, x) = Sum_{k=0..n} 2^(n - k) * Sum_{j=0..k} (x^j * binomial(k, j) * (2*j + 1)^n), (secant case).

1, 3, 3, 7, 36, 25, 15, 297, 625, 343, 31, 2106, 10000, 14406, 6561, 63, 13851, 131250, 369754, 413343, 161051, 127, 87480, 1546875, 7529536, 15411789, 14172488, 4826809, 255, 540189, 17109375, 134237509, 444816117, 721025327, 564736653, 170859375

Offset: 0

Author

Peter Luschny, May 31 2023

Keywords

• nonn
• tabl

Comments

Here we give an inclusion-exclusion representation of 2^n*Euler(n) (see A122045 and A002436), in A363399 we give such a representation for 2^n*Euler(n, 1) = A155585(n), and in A363400 one for the combined sequences.

Examples

			The triangle T(n, k) starts:
  [0]   1;
  [1]   3,      3;
  [2]   7,     36,       25;
  [3]  15,    297,      625,       343;
  [4]  31,   2106,    10000,     14406,      6561;
  [5]  63,  13851,   131250,    369754,    413343,    161051;
  [6] 127,  87480,  1546875,   7529536,  15411789,  14172488,   4826809;
  [7] 255, 540189, 17109375, 134237509, 444816117, 721025327, 564736653, 170859375;
		

Links

• Index entries for sequences related to Euler numbers.

Crossrefs

Cf. A122045 (alternating row sums), A363396 (row sums), A126646 (column 0), A085527 (main diagonal), A141475 (central terms).

Cf. A363399 (tangent case), A363400 (combined case).

Cf. A122045, A002436.

Programs

Maple

P := (n, x) -> add(add(x^j*binomial(k, j)*(2*j + 1)^n, j=0..k)*2^(n-k), k=0..n):
T := (n, k) -> coeff(P(n, x), x, k): seq(seq(T(n, k), k = 0..n), n = 0..7);

Mathematica

(* From Detlef Meya, Oct 04 2023: (Start) *)
T[n_, k_] := (2*k+1)^n*(2^(n+1) - Sum[Binomial[n+1, j], {j,0,k}]);
(* Or: *)
T[n_, k_] := (2*k+1)^n*Binomial[n+1, k+1]*Hypergeometric2F1[1, k-n, k+2, -1];
Flatten[Table[T[n, k], {n, 0, 7}, {k, 0, n}]]  (* End *)

Formula

Sum_{k=0..n} (-1)^k*T(n, k) = 2^n*Euler(n) = 4^n*Euler(n, 1/2).

(Sum_{k=0..n} (-1)^k*T(n, k)) / 2^n = Euler(n) = 2^n*Euler(n, 1/2) = A122045(n).

Sum_{k=0..2*n} (-1)^k*T(2*n, k) = 4^n*Euler(2*n) = 16^n*Euler(2*n, 1/2) = (-1)^n*A002436(n).

From Detlef Meya, Oct 04 2023: (Start)

T(n, k) = (2*k + 1)^n * binomial(n+1, k+1) * hypergeom([1, k-n], [k+2], -1).

T(n, k) = (2*k + 1)^n * (2^(n + 1) - Sum_{j=0..k} binomial(n+1, j)). (End)