A279006 Alternating Jacobsthal triangle read by rows (second version).

1, 1, 1, 1, 0, 1, 1, -1, 1, 1, 1, -2, 2, 0, 1, 1, -3, 4, -2, 1, 1, 1, -4, 7, -6, 3, 0, 1, 1, -5, 11, -13, 9, -3, 1, 1, 1, -6, 16, -24, 22, -12, 4, 0, 1, 1, -7, 22, -40, 46, -34, 16, -4, 1, 1, 1, -8, 29, -62, 86, -80, 50, -20, 5, 0, 1, 1, -9, 37, -91, 148, -166, 130, -70, 25, -5, 1, 1

Offset: 0

Author

N. J. A. Sloane, Dec 07 2016

Keywords

• sign
• tabl

Examples

			Triangle begins:
  1,
  1,   1,
  1,   0,   1,
  1,  -1,   1,   1,
  1,  -2,   2,   0,   1,
  1,  -3,   4,  -2,   1,   1,
  1,  -4,   7,  -6,   3,   0,   1,
  1,  -5,  11, -13,   9,  -3,   1,   1,
  1,  -6,  16, -24,  22, -12,   4,   0,   1,
  ...
		

Links

• Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.

Crossrefs

See A112468, A112555 and A108561 for other versions.

Columns give A000124, A003600, A223718, A257890, A223659.

Programs

Maple

T := (n, k) -> local j; add((-1)^j*binomial(n-k-1+j, j), j = 0..k):
seq(print(seq(T(n, k), k = 0..n)), n = 0..9);  # Peter Luschny, Aug 30 2024

Mathematica

T[i_, i_] = T[, 0] = 1; T[i, j_] := T[i, j] = T[i-1, j] - T[i-1, j-1];
Table[T[i, j], {i, 0, 11}, {j, 0, i}] // Flatten (* Jean-François Alcover, Sep 06 2018 *)
T[n_, k_] := 2^k*Hypergeometric2F1[-n, -k, -k, 1/2]; Table[T[n, k], {n, 0, 11}, {k, 0, n}]//Flatten (* Detlef Meya, Aug 30 2024 *)

PARI

\\ using arxiv (3.1) and (3.7) formulas where A is A220074 and B is this sequence
A(i, j) = if ((i < 0), 0, if (j==0, 1, A(i - 1, j - 1) - A(i - 1, j))); \\ A220074
B(i, j) = A(i, i-j);
tabl(nn) = for (i=0, nn, for (j=0, i, print1(B(i,j), ", ")); print()); \\ Michel Marcus, Jun 17 2017

Formula

T(i, j) = A220074(i, i-j). See (3.7) in arxiv link. - Michel Marcus, Jun 17 2017

T(n, k) = 2^k*hypergeom([-n, -k], [-k], 1/2). - Detlef Meya, Aug 30 2024

Extensions

More terms from Michel Marcus, Jun 17 2017