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A094346
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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.
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0
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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COMMENTS
| Diagonals : A000007, A001469, A005440; A000182, A005990 . Row sums : A001469.
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REFERENCES
| D. Dumont, Sur une conjecture de Gandhi concernant les nombres de Genocchi. Discrete Mathematics 1(1972), 321-327.
D. Dumont, Interpretations combinatoires des nombres de Genocchi, Duke Math.J., 41(2)(1974), 305-318.
J. M. Gandhi, A conjectured representation of Genocchi numbers, Amer. Math. Monthly, 77(1)(1970), 505-506.
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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) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 10 2004
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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 ;...
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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 */
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CROSSREFS
| Cf. A094665 A083061.
Sequence in context: A127160 A131330 A020826 * A074104 A071411 A121065
Adjacent sequences: A094343 A094344 A094345 * A094347 A094348 A094349
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KEYWORD
| nonn,tabl
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AUTHOR
| DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 08 2004, Jun 13 2007
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