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A156368 A ménage triangle. 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,9
REFERENCES
A. Kaufmann, Introduction à la combinatorique en vue des applications, p.188-189, Dunod, Paris, 1968. - Philippe Deléham, Apr 04 2014
LINKS
FORMULA
T(n, k) = Sum_{j=0..n} (-1)^(k+j)*binomial(j, k)*binomial(2*n-j, j)*(n-j)!.
T(n, 0) = A000271(n).
Sum_{k=0..n} T(n, k) = n!.
Equals A155856*A007318^{-1}.
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
EXAMPLE
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;
MATHEMATICA
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 *)
PROG
(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) )
flatten([[A156368(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 05 2021
CROSSREFS
Sequence in context: A352794 A276228 A188938 * A240665 A068958 A238106
KEYWORD
easy,nonn,tabl
AUTHOR
Paul Barry, Feb 08 2009
STATUS
approved

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Last modified March 29 06:15 EDT 2024. Contains 371265 sequences. (Running on oeis4.)