A363399 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) * (j + 1)^n), (tangent case).

1, 3, 2, 7, 16, 9, 15, 88, 135, 64, 31, 416, 1296, 1536, 625, 63, 1824, 10206, 22528, 21875, 7776, 127, 7680, 72171, 262144, 453125, 373248, 117649, 255, 31616, 478953, 2670592, 7265625, 10357632, 7411887, 2097152, 511, 128512, 3057426, 25034752, 100000000, 218350080, 265180846, 167772160, 43046721

Offset: 0

Author

Peter Luschny, May 31 2023

Keywords

• nonn
• tabl

Comments

Here we give an inclusion-exclusion representation of 2^n*Euler(n, 1) = A155585(n), in A363398 we give such a representation for 2^n*Euler(n), and in A363400 one for the combined sequences.

Examples

			The triangle T(n, k) begins:
  [0]   1;
  [1]   3,     2;
  [2]   7,    16,      9;
  [3]  15,    88,    135,      64;
  [4]  31,   416,   1296,    1536,     625;
  [5]  63,  1824,  10206,   22528,   21875,     7776;
  [6] 127,  7680,  72171,  262144,  453125,   373248,  117649;
  [7] 255, 31616, 478953, 2670592, 7265625, 10357632, 7411887, 2097152;
		

Links

• Index entries for sequences related to Euler numbers.

Crossrefs

Cf. A155585 (alternating row sums), A363397 (row sums), A126646 (column 0), A000169 (main diagonal), A163395 (central terms), A084623.

Cf. A363398 (secant case), A363400 (combined case).

Programs

Maple

P := (n, x) -> add(add(x^j*binomial(k, j)*(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..8);

Mathematica

(* From  Detlef Meya, Oct 04 2023: (Start) *)
T[n_, k_] := (k+1)^n*(2^(n+1)-Sum[Binomial[n+1, j], {j, 0, k}]);
(* Or *)
T[n_, k_] := (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, 1) = (-2)^n*Euler(n, 0) = A155585(n).

From Detlef Meya, Oct 04 2023: (Start)

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

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