OFFSET
0,1
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).
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
FORMULA
T(n, k) = 2*n*binomial(2*n-k, k)*(n-k)!/(2*n-k), with T(0, 0) = 2.
EXAMPLE
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;
...
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 20 2009
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
STATUS
approved