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A054655
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Triangle T(n,k) giving coefficients in expansion of n!*binomial(x-n,n) in powers of x.
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6
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1, 1, -1, 1, -5, 6, 1, -12, 47, -60, 1, -22, 179, -638, 840, 1, -35, 485, -3325, 11274, -15120, 1, -51, 1075, -11985, 74524, -245004, 332640, 1, -70, 2086, -34300, 336049, -1961470, 6314664, -8648640, 1, -92, 3682, -83720, 1182769
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OFFSET
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0,5
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LINKS
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FORMULA
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n!*binomial(x-n, n) = Sum_{k=0..n} T(n, k)*x^(n-k).
G.f.: Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*y^k = hypergeom([1, -1/(2*y), (1/2)*(-1+y)/y], [-1/y], -4*x*y).
E.g.f.: Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*y^k/n! = (1+4*x*y)^(-1/2)*((1+sqrt(1+4*x*y))/2)^(1+1/y). (End)
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EXAMPLE
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Triangle begins:
1;
1, -1;
1, -5, 6;
1, -12, 47, -60;
1, -22, 179, -638, 840;
1, -35, 485, -3325, 11274, -15120;
1, -51, 1075, -11985, 74524, -245004, 332640;
1, -70, 2086, -34300, 336049, -1961470, 6314664, -8648640;
...
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MAPLE
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a054655_row := proc(n) local k; seq(coeff(expand((-1)^n*pochhammer (n-x, n)), x, n-k), k=0..n) end: # Peter Luschny, Nov 28 2010
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MATHEMATICA
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row[n_] := Table[ Coefficient[(-1)^n*Pochhammer[n - x, n], x, n - k], {k, 0, n}]; A054655 = Flatten[ Table[ row[n], {n, 0, 8}]] (* Jean-François Alcover, Apr 06 2012, after Maple *)
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PROG
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(PARI) T(n, k)=polcoef(n!*binomial(x-n, n), n-k);
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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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