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A156995 Triangle array read by rows: a(0,0) = 2 and for n>=1, 0<=m<=n, a(n,m) = binomial(2*n-m,m)*(n-m)!*2*n/(2*n-m). 3
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_{m=0..n} x^m * binomial(2*n - m, m) * (n - m)! * 2 * n / (2*n - m). 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

Table of n, a(n) for n=0..50.

FORMULA

a(0,0) = 2.

For n>=1, a(n,m) = binomial(2*n-m,m)*(n-m)!*2*n/(2*n-m).

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

Table[CoefficientList[If[n == 0, 2, Sum[Binomial[2*n - m, m]*(n - m)!*( 2*n/(2*n - m))x^m, {m, 0, n}]], x], {n, 0, 12}];

Flatten[%]

CROSSREFS

Row sums are A300484.

Sequence in context: A308302 A225530 A020475 * A131183 A133770 A288310

Adjacent sequences:  A156992 A156993 A156994 * A156996 A156997 A156998

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Feb 20 2009

EXTENSIONS

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

STATUS

approved

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Last modified January 26 14:08 EST 2020. Contains 331280 sequences. (Running on oeis4.)