OFFSET
0,2
COMMENTS
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
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, May 31 2023
STATUS
approved