

A156995


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


5



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
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OFFSET

0,1


COMMENTS

For n>=1, o.g.f. of nth row is a polynomial p(x,n) = Sum_{k=0..n} ( 2*n*(nk)! * binomial(2*nk, k)/(2*nk)) * 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. 197199


LINKS



FORMULA

T(n, k) = 2*n*binomial(2*nk, k)*(nk)!/(2*nk), 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*nk, k]*(nk)!/(2*nk)];
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(nk)*Binomial(2*nk, k)/(2*nk) >;
(Sage)
def A156995(n, k): return 2 if (k==n) else 2*n*factorial(nk)*binomial(2*nk, k)/(2*nk)


CROSSREFS



KEYWORD



AUTHOR



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



