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A062190
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Coefficient triangle of certain polynomials N(5; m,x).
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29
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1, 1, 6, 1, 14, 21, 1, 24, 84, 56, 1, 36, 216, 336, 126, 1, 50, 450, 1200, 1050, 252, 1, 66, 825, 3300, 4950, 2772, 462, 1, 84, 1386, 7700, 17325, 16632, 6468, 792, 1, 104, 2184, 16016, 50050, 72072, 48048, 13728
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OFFSET
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0,3
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COMMENTS
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The e.g.f. of the m-th (unsigned) column sequence without leading zeros of the generalized (a=5) Laguerre triangle L(5; n+m,m)= A062138(n+m,m), n >= 0, is N(5; m,x)/(1-x)^(2*(m+3)), with the row polynomials N(5; m,x) := Sum_{k=0..m} a(m,k)*x^k.
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LINKS
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FORMULA
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a(m, k) = [x^k]N(5; m, x), with N(5; m, x) = ((1-x)^(2*(m+3)))*(d^m/dx^m)(x^m/(m!*(1-x)^(m+6))).
N(5; m, x) = Sum_{j=0..m} ((binomial(m, j)*(2*m+5-j)!/((m+5)!*(m-j)!))*(x^(m-j))*(1-x)^j).
N(5; m, x)= x^m*(2*m+5)! * 2F1(-m, -m; -2*m-5; (x-1)/x)/((m+5)!*m!). [Jean-François Alcover, Sep 18 2013]
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EXAMPLE
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1,
1, 6,
1, 14, 21,
1, 24, 84, 56,
1, 36, 216, 336, 126,
1, 50, 450, 1200, 1050, 252,
1, 66, 825, 3300, 4950, 2772, 462,
1, 84, 1386, 7700, 17325, 16632, 6468, 792,
1, 104, 2184, 16016, 50050, 72072, 48048, 13728, 1287,
1, 126, 3276, 30576, 126126, 252252, 252252, 123552, 27027, 2002,
1, 150, 4725, 54600, 286650, 756756, 1051050, 772200, 289575, 50050, ...
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MAPLE
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add( (binomial(m, j)*(2*m+5-j)!/((m+5)!*(m-j)!))*(x^(m-j))*(1-x)^j, j=0..m) ;
coeftayl(%, x=0, k) ;
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MATHEMATICA
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NN[5, m_, x_] := x^m*(2*m+5)!*Hypergeometric2F1[-m, -m, -2*m-5, (x-1)/x]/((m+5)!*m!); Table[CoefficientList[NN[5, m, x], x], {m, 0, 8}] // Flatten (* Jean-François Alcover, Sep 18 2013 *)
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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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