OFFSET
0,6
LINKS
D. Dumont, Sur une conjecture de Gandhi concernant les nombres de Genocchi, Discrete Mathematics 1 (1972) 321-327.
D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.
J. M. Gandhi, Research Problems: A Conjectured Representation of Genocchi Numbers, Amer. Math. Monthly 77 (1970), no. 5, 505-506. MR1535914
Andrei K. Svinin, Tuenter polynomials and a Catalan triangle, arXiv:1603.05748 [math.CO], 2016. (Has a signed version of this triangle, see p. 1).
FORMULA
For n>=1, Sum_{k =1..n} T(n, k)*x^(k-1) = G(x, n), n-th Gandhi polynomial; the Gandhi polynomials are defined by G(x, n) = (x+1)^2*G(x+1, n-1) - x^2*G(x, n-1), G(x, 1) = 1. Sum_{k =0..n} T(n, k)*2^(2n-k) = A000182(n+1), tangent numbers. Sum_{k =0..n} T(n, k) = A001469(n+1), Genocchi numbers of first kind.
Sum_{k = 0..n} T(n, k)*2^(n-k) = A002105(n+1). - Philippe Deléham, Jun 10 2004
EXAMPLE
Triangle begins:
1;
0, 1;
0, 1, 2;
0, 3, 8, 6;
0, 17, 54, 60, 24;
0, 155, 556, 762, 480, 120;
0, 2073, 8146, 12840, 10248, 4200, 720;
0, 38227, 161424, 282078, 263040, 139440, 40320, 5040;
0, 929569, 4163438, 7886580, 8240952, 5170800, 1965600, 423360, 40320; ...
MATHEMATICA
G[_, 1] = 1;
G[x_, n_] := G[x, n] = (x+1)^2 G[x+1, n-1] - x^2 G[x, n-1] // Expand;
row[0] = {1};
row[n_] := CoefficientList[x G[x, n], x];
Table[row[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Aug 17 2018 *)
PROG
(PARI) {T(n, k) = local( A = x); if( k<0 || k>n, 0, for( j = 1, n, A = x^2 * ( subst(A, x, x+1) - A)); polcoeff( A, k+1))} /* Michael Somos, Apr 10 2011 */
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Philippe Deléham, Jun 08 2004, Jun 13 2007
STATUS
approved