A156995 Triangle T(n, k) = 2*n*binomial(2*n-k, k)*(n-k)!/(2*n-k), with T(0, 0) = 2, read by rows.

2, 1, 2, 2, 4, 2, 6, 12, 9, 2, 24, 48, 40, 16, 2, 120, 240, 210, 100, 25, 2, 720, 1440, 1296, 672, 210, 36, 2, 5040, 10080, 9240, 5040, 1764, 392, 49, 2, 40320, 80640, 74880, 42240, 15840, 4032, 672, 64, 2, 362880, 725760, 680400, 393120, 154440, 42768

Offset: 0

Author

Roger L. Bagula, Feb 20 2009

Keywords

• nonn
• tabl

Comments

For n>=1, o.g.f. of n-th row is a polynomial p(x,n) = Sum_{k=0..n} ( 2*n*(n-k)! * binomial(2*n-k, k)/(2*n-k)) * x^k. These polynomials are hit polynomials for the reduced ménage problem (Riordan 1958).

Examples

			Triangle starts with:
  n=0:     2;
  n=1:     1,     2;
  n=2:     2,     4,     2;
  n=3:     6,    12,     9,     2;
  n=4:    24,    48,    40,    16,     2;
  n=5:   120,   240,   210,   100,    25,    2;
  n=6:   720,  1440,  1296,   672,   210,   36,   2;
  n=7:  5040, 10080,  9240,  5040,  1764,  392,  49,  2;
  n=8: 40320, 80640, 74880, 42240, 15840, 4032, 672, 64, 2;
  ...
		

References

• J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 197-199

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Row sums are A300484.

Programs

Magma

A156995:= func< n,k | n eq 0 select 2 else 2*n*Factorial(n-k)*Binomial(2*n-k, k)/(2*n-k) >;
[A156995(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 14 2021

Mathematica

T[n_, k_]:= If[n==0, 2, 2*n*Binomial[2*n-k, k]*(n-k)!/(2*n-k)];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, May 14 2021 *)

Sage

def A156995(n,k): return 2 if (k==n) else 2*n*factorial(n-k)*binomial(2*n-k,k)/(2*n-k)
flatten([[A156995(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 14 2021

Formula

T(n, k) = 2*n*binomial(2*n-k, k)*(n-k)!/(2*n-k), with T(0, 0) = 2.

Extensions

Edited and changed T(0,0) = 2 (to make formula continuous and constant along the diagonal k = n) by Max Alekseyev, Mar 06 2018