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A086915
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Triangle read by rows: T(n,k) = 2^k * (n!/k!)*binomial(n-1,k-1).
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3
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2, 4, 4, 12, 24, 8, 48, 144, 96, 16, 240, 960, 960, 320, 32, 1440, 7200, 9600, 4800, 960, 64, 10080, 60480, 100800, 67200, 20160, 2688, 128, 80640, 564480, 1128960, 940800, 376320, 75264, 7168, 256, 725760, 5806080, 13547520, 13547520, 6773760, 1806336
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OFFSET
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1,1
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COMMENTS
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The coefficients of n! * L_n(-2*x,-1), where n! * L_n(-x,-1) are the normalized, unsigned Laguerre polynomials of order -1 of A105278, also known as the Lah polynomials, which are also a shifted version of n! * L_n(-x,1). Cf. p. 8 of the Gross and Matytsin link. - Tom Copeland, Sep 30 2016
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LINKS
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FORMULA
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E.g.f.: exp(2*x*y/(1-x)).
Sum_{k=1..n} T(n, k) = 2 * n! * Hypergeometric1F1([1-n], [2], -2) = 2*(n-1)! * LaguerreL(n-1, 1, -2) = A253286(n, 2). (End)
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EXAMPLE
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Triangle begins:
2;
4, 4;
12, 24, 8;
48, 144, 96, 16;
...
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MAPLE
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# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ...) as column 0.
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MATHEMATICA
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Flatten[Table[n!/k! Binomial[n-1, k-1]2^k, {n, 10}, {k, n}]] (* Harvey P. Dale, May 25 2011 *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[2*(#+1)!&, rows = 12];
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PROG
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(PARI) for(n=1, 10, for(k=1, n, print1(n!/k!*binomial(n-1, k-1)*2^k, ", "))) \\ G. C. Greubel, May 23 2018
(Magma) [Factorial(n)*Binomial(n-1, k-1)*2^k/Factorial(k): k in [1..n], n in [1..10]]; // G. C. Greubel, May 23 2018
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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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