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

Author

Paul Barry, Feb 08 2009

Keywords

• easy
• nonn
• tabl

Examples

			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;
		

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

Crossrefs

Cf. A000142, A000271, A007318, A155856.

Programs

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 *)

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

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