A156368 A ménage triangle.

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

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

Alois P. Heinz, Rows n = 0..140, flattened

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

Cf. A000142, A000271, A007318, A155856.

Sequence in context: A352794 A276228 A188938 * A240665 A068958 A238106

Adjacent sequences: A156365 A156366 A156367 * A156369 A156370 A156371

Keyword

easy,nonn,tabl

Author

Paul Barry, Feb 08 2009

Status

approved