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

0,2

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.

Links

Table of n, a(n) for n=0..35.

Index entries for sequences related to Euler numbers.

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..2n}((-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)

Example

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;

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 *)

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.

Sequence in context: A173321 A191498 A065747 * A232309 A273043 A066142

Adjacent sequences: A363395 A363396 A363397 * A363399 A363400 A363401

Keyword

nonn,tabl

Author

Peter Luschny, May 31 2023

Status

approved