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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
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OFFSET
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0,6
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COMMENTS
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LINKS
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FORMULA
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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.
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EXAMPLE
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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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MATHEMATICA
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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];
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PROG
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(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
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KEYWORD
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AUTHOR
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STATUS
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approved
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