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A156368
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A ménage triangle.
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1
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1, 0, 1, 0, 1, 1, 1, 1, 3, 1, 3, 8, 6, 6, 1, 16, 35, 38, 20, 10, 1, 96, 211, 213, 134, 50, 15, 1, 675, 1459, 1479, 915, 385, 105, 21, 1, 5413, 11584, 11692, 7324, 3130, 952, 196, 28, 1, 48800, 103605, 104364, 65784, 28764, 9090, 2100, 336, 36, 1
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history;
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OFFSET
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0,9
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REFERENCES
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A. Kaufmann, Introduction à la combinatorique en vue des applications, p.188-189, Dunod, Paris, 1968. - Philippe Deléham, Apr 04 2014
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LINKS
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FORMULA
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T(n, k) = Sum_{j=0..n} (-1)^(k+j)*binomial(j, k)*binomial(2*n-j, j)*(n-j)!.
Sum_{k=0..n} T(n, k) = n!.
G.f.: 1/(1 +x -x*y -x/(1 +x -x*y -x/(1 +x -x*y -2*x/(1 +x -x*y -2*x/(1 +x -x*y -3*x/(1 +x -x*y -3*x/(1 +x -x*y -4*x/(1 + ... (continued fraction).
G.f.: Sum_{n>=0} n! * x^n/(1 + (1-y)*x)^(2*n+1). - Ira M. Gessel, Jan 15 2013
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EXAMPLE
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Triangle begins:
1;
0, 1;
0, 1, 1;
1, 1, 3, 1;
3, 8, 6, 6, 1;
16, 35, 38, 20, 10, 1;
96, 211, 213, 134, 50, 15, 1;
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MATHEMATICA
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T[n_, k_]:= Sum[(-1)^(k+j)*Binomial[j, k]*Binomial[2*n-j, j]*(n-j)!, {j, 0, n}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 05 2021 *)
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PROG
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(Sage)
def A156368(n, k): return sum( (-1)^(k+j)*binomial(j, k)*binomial(2*n-j, j)*factorial(n-j) for j in (0..n) )
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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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