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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).
3
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.
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
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, May 31 2023
STATUS
approved