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A257635
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Triangle with n-th row polynomial equal to Product_{k = 1..n} (x + n + k).
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2
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1, 2, 1, 12, 7, 1, 120, 74, 15, 1, 1680, 1066, 251, 26, 1, 30240, 19524, 5000, 635, 40, 1, 665280, 434568, 117454, 16815, 1345, 57, 1, 17297280, 11393808, 3197348, 495544, 45815, 2527, 77, 1, 518918400, 343976400, 99236556, 16275700, 1659889, 107800, 4354, 100, 1
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OFFSET
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0,2
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COMMENTS
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The row polynomials are a Sheffer sequence. For the associated polynomial sequence of binomial type see A038455.
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LINKS
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FORMULA
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E.g.f.: A(x,t) = B(t)*C(t)^x = 1 + (2 + x)*t + (3 + x)*(4 + x)*t^2/2! + (4 + x)*(5 + x)*(6 + x)*t^3/3! + ..., where B(t) = 1/sqrt(1 - 4*t) is the o.g.f. for A000984 and C(t) = (1 - sqrt(1 - 4*t))/(2*t) is the o.g.f. for A000108.
n-th row polynomial: n!*binomial(2*n + x,n).
T(n, k) = (-1)^k*n!*[x^k] hypergeom([-n, -x + n], [-n], 1). - Peter Luschny, Nov 27 2021
T(n, k) = [x^k] Gamma(2*n + x + 1) / Gamma(n + x + 1). - Peter Luschny, Mar 21 2022
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EXAMPLE
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Triangle begins:
[0] 1;
[1] 2, 1;
[2] 12, 7, 1;
[3] 120, 74, 15, 1;
[4] 1680, 1066, 251, 26, 1;
[5] 30240, 19524, 5000, 635, 40, 1;
[6] 665280, 434568, 117454, 16815, 1345, 57, 1;
...
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MAPLE
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seq(seq(coeff(product(n + x + k, k = 1 .. n), x, i), i = 0..n), n = 0..8);
# Alternative:
p := n -> n!*hypergeom([-n, -x + n], [-n], 1):
seq(seq((-1)^k*coeff(simplify(p(n)), x, k), k=0..n), n=0..6); # Peter Luschny, Nov 27 2021
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MATHEMATICA
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p[n_, x_] := FunctionExpand[Gamma[2*n + x + 1] / Gamma[n + x + 1]];
Table[CoefficientList[p[n, x], x], {n, 0, 8}] // Flatten (* Peter Luschny, Mar 21 2022 *)
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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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