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

%I A279006 #46 Jun 23 2025 11:06:39
%S A279006 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,
%T A279006 11,-13,9,-3,1,1,1,-6,16,-24,22,-12,4,0,1,1,-7,22,-40,46,-34,16,-4,1,
%U A279006 1,1,-8,29,-62,86,-80,50,-20,5,0,1,1,-9,37,-91,148,-166,130,-70,25,-5,1,1
%N A279006 Alternating Jacobsthal triangle read by rows (second version).
%H A279006 Kyu-Hwan Lee and Se-jin Oh, <a href="http://arxiv.org/abs/1601.06685">Catalan triangle numbers and binomial coefficients</a>, arXiv:1601.06685 [math.CO], 2016.
%F A279006 T(i, j) = A220074(i, i-j). See (3.7) in arxiv link. - _Michel Marcus_, Jun 17 2017
%F A279006 T(n, k) = 2^k*hypergeom([-n, -k], [-k], 1/2). - _Detlef Meya_, Aug 30 2024
%e A279006 Triangle begins:
%e A279006   1,
%e A279006   1,   1,
%e A279006   1,   0,   1,
%e A279006   1,  -1,   1,   1,
%e A279006   1,  -2,   2,   0,   1,
%e A279006   1,  -3,   4,  -2,   1,   1,
%e A279006   1,  -4,   7,  -6,   3,   0,   1,
%e A279006   1,  -5,  11, -13,   9,  -3,   1,   1,
%e A279006   1,  -6,  16, -24,  22, -12,   4,   0,   1,
%e A279006   ...
%p A279006 T := (n, k) -> local j; add((-1)^j*binomial(n-k-1+j, j), j = 0..k):
%p A279006 seq(print(seq(T(n, k), k = 0..n)), n = 0..9);  # _Peter Luschny_, Aug 30 2024
%t A279006 T[i_, i_] = T[_, 0] = 1; T[i_, j_] := T[i, j] = T[i-1, j] - T[i-1, j-1];
%t A279006 Table[T[i, j], {i, 0, 11}, {j, 0, i}] // Flatten (* _Jean-François Alcover_, Sep 06 2018 *)
%t A279006 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 *)
%o A279006 (PARI) \\ using arxiv (3.1) and (3.7) formulas where A is A220074 and B is this sequence
%o A279006 A(i, j) = if ((i < 0), 0, if (j==0, 1, A(i - 1, j - 1) - A(i - 1, j))); \\ A220074
%o A279006 B(i, j) = A(i, i-j);
%o A279006 tabl(nn) = for (i=0, nn, for (j=0, i, print1(B(i,j), ", ")); print()); \\ _Michel Marcus_, Jun 17 2017
%Y A279006 See A112468, A112555 and A108561 for other versions.
%Y A279006 Columns give A000124, A003600, A223718, A257890, A223659.
%K A279006 sign,tabl
%O A279006 0,12
%A A279006 _N. J. A. Sloane_, Dec 07 2016
%E A279006 More terms from _Michel Marcus_, Jun 17 2017