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A123516
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Triangle read by rows: T(n,k) = (-1)^k * n! * 2^(n-2*k) * binomial(n,k) * binomial(2*k,k) (0<=k<=n).
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2
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1, 2, -1, 8, -8, 3, 48, -72, 54, -15, 384, -768, 864, -480, 105, 3840, -9600, 14400, -12000, 5250, -945, 46080, -138240, 259200, -288000, 189000, -68040, 10395, 645120, -2257920, 5080320, -7056000, 6174000, -3333960, 1018710, -135135, 10321920, -41287680, 108380160, -180633600, 197568000
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OFFSET
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0,2
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COMMENTS
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Row sums yield the double factorial numbers (A001147).
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REFERENCES
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B. T. Gill, Math. Magazine, vol. 79, No. 4, 2006, p. 313, problem 1729.
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LINKS
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FORMULA
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EXAMPLE
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Triangle begins:
1;
2, -1;
8, -8, 3;
48, -72, 54, -15;
384, -768, 864, -480, 105;
3840, -9600, 14400, -12000, 5250, -945;
46080, -138240, 259200, -288000, 189000, -68040, 10395;
...
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MAPLE
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T:=(n, k)->(-1)^k*n!*2^(n-2*k)*binomial(n, k)*binomial(2*k, k): for n from 0 to 8 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
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MATHEMATICA
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Table[(-1)^k*n! 2^(n - 2 k)*Binomial[n, k]*Binomial[2*k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Oct 14 2017 *)
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PROG
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(PARI) for(n=0, 10, for(k=0, n, print1((-1)^k*n!*2^(n-2*k)*binomial(n, k)* binomial(2*k, k), ", "))) \\ G. C. Greubel, Oct 14 2017
(Magma) /* As triangle * / [[(-1)^k*Factorial(n)*2^(n-2*k)* Binomial(n, k)*Binomial(2*k, k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Oct 15 2017
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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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