A156368 - OEIS

%I #20 Jun 05 2021 16:46:26

%S 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,

%T 1,675,1459,1479,915,385,105,21,1,5413,11584,11692,7324,3130,952,196,

%U 28,1,48800,103605,104364,65784,28764,9090,2100,336,36,1

%N A ménage triangle.

%D A. Kaufmann, Introduction à la combinatorique en vue des applications, p.188-189, Dunod, Paris, 1968. - _Philippe Deléham_, Apr 04 2014

%H Alois P. Heinz, <a href="/A156368/b156368.txt">Rows n = 0..140, flattened</a>

%F T(n, k) = Sum_{j=0..n} (-1)^(k+j)*binomial(j, k)*binomial(2*n-j, j)*(n-j)!.

%F T(n, 0) = A000271(n).

%F Sum_{k=0..n} T(n, k) = n!.

%F Equals A155856*A007318^{-1}.

%F 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).

%F G.f.: Sum_{n>=0} n! * x^n/(1 + (1-y)*x)^(2*n+1). - _Ira M. Gessel_, Jan 15 2013

%e Triangle begins:

%e    1;

%e    0,   1;

%e    0,   1,   1;

%e    1,   1,   3,   1;

%e    3,   8,   6,   6,  1;

%e   16,  35,  38,  20, 10,  1;

%e   96, 211, 213, 134, 50, 15,  1;

%t T[n_,k_]:= Sum[(-1)^(k+j)*Binomial[j, k]*Binomial[2*n-j, j]*(n-j)!, {j,0,n}];

%t Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 05 2021 *)

%o (Sage)

%o 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) )

%o flatten([[A156368(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 05 2021

%Y Cf. A000142, A000271, A007318, A155856.

%K easy,nonn,tabl

%O 0,9

%A _Paul Barry_, Feb 08 2009