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

0,2

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.

Links

Table of n, a(n) for n=0..44.

Index entries for sequences related to Euler numbers.

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)

Example

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;

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 *)

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).

Sequence in context: A344494 A286940 A049968 * A049970 A344211 A104528

Adjacent sequences: A363396 A363397 A363398 * A363400 A363401 A363402

Keyword

nonn,tabl

Author

Peter Luschny, May 31 2023

Status

approved