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A156995
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Triangle T(n, k) = 2*n*binomial(2*n-k, k)*(n-k)!/(2*n-k), with T(0, 0) = 2, read by rows.
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5
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2, 1, 2, 2, 4, 2, 6, 12, 9, 2, 24, 48, 40, 16, 2, 120, 240, 210, 100, 25, 2, 720, 1440, 1296, 672, 210, 36, 2, 5040, 10080, 9240, 5040, 1764, 392, 49, 2, 40320, 80640, 74880, 42240, 15840, 4032, 672, 64, 2, 362880, 725760, 680400, 393120, 154440, 42768
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OFFSET
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0,1
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COMMENTS
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For n>=1, o.g.f. of n-th row is a polynomial p(x,n) = Sum_{k=0..n} ( 2*n*(n-k)! * binomial(2*n-k, k)/(2*n-k)) * x^k. These polynomials are hit polynomials for the reduced ménage problem (Riordan 1958).
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REFERENCES
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J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 197-199
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LINKS
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FORMULA
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T(n, k) = 2*n*binomial(2*n-k, k)*(n-k)!/(2*n-k), with T(0, 0) = 2.
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EXAMPLE
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Triangle starts with:
n=0: 2;
n=1: 1, 2;
n=2: 2, 4, 2;
n=3: 6, 12, 9, 2;
n=4: 24, 48, 40, 16, 2;
n=5: 120, 240, 210, 100, 25, 2;
n=6: 720, 1440, 1296, 672, 210, 36, 2;
n=7: 5040, 10080, 9240, 5040, 1764, 392, 49, 2;
n=8: 40320, 80640, 74880, 42240, 15840, 4032, 672, 64, 2;
...
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MATHEMATICA
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T[n_, k_]:= If[n==0, 2, 2*n*Binomial[2*n-k, k]*(n-k)!/(2*n-k)];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, May 14 2021 *)
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PROG
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(Magma)
A156995:= func< n, k | n eq 0 select 2 else 2*n*Factorial(n-k)*Binomial(2*n-k, k)/(2*n-k) >;
(Sage)
def A156995(n, k): return 2 if (k==n) else 2*n*factorial(n-k)*binomial(2*n-k, k)/(2*n-k)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Edited and changed T(0,0) = 2 (to make formula continuous and constant along the diagonal k = n) by Max Alekseyev, Mar 06 2018
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STATUS
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approved
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