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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. 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 (list; table; graph; refs; listen; history; text; internal format)
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

Row sums are A300484.

Sequence in context: A308302 A225530 A020475 * A131183 A346063 A133770

Adjacent sequences:  A156992 A156993 A156994 * A156996 A156997 A156998

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

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Last modified August 7 20:58 EDT 2022. Contains 355994 sequences. (Running on oeis4.)