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A062145
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Triangle read by rows. T{n, k] = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.
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21
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1, 1, 4, 1, 10, 10, 1, 18, 45, 20, 1, 28, 126, 140, 35, 1, 40, 280, 560, 350, 56, 1, 54, 540, 1680, 1890, 756, 84, 1, 70, 945, 4200, 7350, 5292, 1470, 120, 1, 88, 1540, 9240, 23100, 25872, 12936, 2640, 165, 1, 108
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OFFSET
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0,3
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COMMENTS
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Coefficient triangle of certain polynomials N(3; m,x).
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LINKS
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FORMULA
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The e.g.f. of the m-th (unsigned) column sequence without leading zeros of the generalized (a=3) Laguerre triangle L(3; n+m, m) = A062137(n+m, m), n >= 0, is N(3; m, x)/(1-x)^(2*(m+2)), with the row polynomials N(3; m, x) := Sum_{k=0..m} a(m, k)*x^k.
N(3; m, x) := ((1-x)^(2*(m+2)))*(d^m/dx^m)(x^m/(m!*(1-x)^(m+4))); a(m, k) = [x^k]N(3; m, x).
N(3; m, x) = Sum_{j=0..m} ((binomial(m, j)*(2*m+3-j)!/((m+3)!*(m-j)!))*(x^(m-j))*(1-x)^j).
N(3; m, x)= x^m*(2*m+3)! * 2F1(-m, -m; -2*m-3; (x-1)/x)/((m+3)!*m!). [Jean-François Alcover, Sep 18 2013]
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EXAMPLE
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As a square array:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
4, 10, 18, 28, 40, 54, 70, 88, ...
10, 45, 126, 280, 540, 945, 1540, ...
20, 140, 560, 1680, 4200, 9240, ...
35, 350, 1890, 7350, 23100, ...
56, 756, 5292, 25872, ...
... (End)
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MATHEMATICA
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NN[3, m_, x_] := x^m*(2*m+3)!*Hypergeometric2F1[-m, -m, -2*m-3, (x-1)/x]/((m+3)!*m!); Table[CoefficientList[NN[3, m, x], x], {m, 0, 9}] // Flatten (* Jean-François Alcover, Sep 18 2013 *)
P[c_, n_, z_] := Sum[Binomial[n, k] Pochhammer[n - k + c, k] z^k / k!, {k, 0, n}];
CL[c_] := Table[CoefficientList[P[c, n, z], z], {n, 0, 5}] // TableForm
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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