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%I #25 Aug 17 2018 10:51:37
%S 1,0,1,0,1,2,0,3,8,6,0,17,54,60,24,0,155,556,762,480,120,0,2073,8146,
%T 12840,10248,4200,720,0,38227,161424,282078,263040,139440,40320,5040,
%U 0,929569,4163438,7886580,8240952,5170800,1965600,423360,40320
%N Another version of triangular array in A036970: triangle T(n,k), 0<=k<=n, read by rows; given by [0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, ...] where DELTA is the operator defined in A084938.
%C Diagonals: A000007, A001469, A005440; A000182, A005990. Row sums: A001469.
%H D. Dumont, <a href="http://dx.doi.org/10.1016/0012-365X(72)90039-8">Sur une conjecture de Gandhi concernant les nombres de Genocchi</a>, Discrete Mathematics 1 (1972) 321-327.
%H D. Dumont, <a href="http://dx.doi.org/10.1215/S0012-7094-74-04134-9">Interprétations combinatoires des nombres de Genocchi</a>, Duke Math. J., 41 (1974), 305-318.
%H J. M. Gandhi, <a href="http://www.jstor.org/stable/2317385">Research Problems: A Conjectured Representation of Genocchi Numbers</a>, Amer. Math. Monthly 77 (1970), no. 5, 505-506. MR1535914
%H Andrei K. Svinin, <a href="http://arxiv.org/abs/1603.05748">Tuenter polynomials and a Catalan triangle</a>, arXiv:1603.05748 [math.CO], 2016. (Has a signed version of this triangle, see p. 1).
%F 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.
%F Sum_{k = 0..n} T(n, k)*2^(n-k) = A002105(n+1). - _Philippe Deléham_, Jun 10 2004
%e Triangle begins:
%e 1;
%e 0, 1;
%e 0, 1, 2;
%e 0, 3, 8, 6;
%e 0, 17, 54, 60, 24;
%e 0, 155, 556, 762, 480, 120;
%e 0, 2073, 8146, 12840, 10248, 4200, 720;
%e 0, 38227, 161424, 282078, 263040, 139440, 40320, 5040;
%e 0, 929569, 4163438, 7886580, 8240952, 5170800, 1965600, 423360, 40320; ...
%t G[_, 1] = 1;
%t G[x_, n_] := G[x, n] = (x+1)^2 G[x+1, n-1] - x^2 G[x, n-1] // Expand;
%t row[0] = {1};
%t row[n_] := CoefficientList[x G[x, n], x];
%t Table[row[n], {n, 0, 8}] // Flatten (* _Jean-François Alcover_, Aug 17 2018 *)
%o (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 */
%Y Cf. A094665, A083061.
%K nonn,tabl
%O 0,6
%A _Philippe Deléham_, Jun 08 2004, Jun 13 2007
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