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A172089 Triangle T(n,m) = n!/(m!!*(n-m)!!) read by rows, where (.)!! = A006882(.) are double factorials. 1
1, 1, 1, 1, 2, 1, 2, 3, 3, 2, 3, 8, 6, 8, 3, 8, 15, 20, 20, 15, 8, 15, 48, 45, 80, 45, 48, 15, 48, 105, 168, 210, 210, 168, 105, 48, 105, 384, 420, 896, 630, 896, 420, 384, 105, 384, 945, 1728, 2520, 3024, 3024, 2520, 1728, 945, 384, 945, 3840, 4725, 11520, 9450, 16128, 9450, 11520, 4725, 3840, 945 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are {1, 2, 4, 10, 28, 86, 296, 1062, 4240, 17202, 77088, ...}.

LINKS

G. C. Greubel, Rows n = 0..100 of triangle, flattened

FORMULA

T(n,m) = A000142(n)/(A006882(m)*A006882(n-m)).

EXAMPLE

Triangle begins

    1;

    1,    1;

    1,    2,    1;

    2,    3,    3,     2;

    3,    8,    6,     8,    3;

    8,   15,   20,    20,   15,     8;

   15,   48,   45,    80,   45,    48,   15;

   48,  105,  168,   210,  210,   168,  105,    48;

  105,  384,  420,   896,  630,   896,  420,   384,  105;

  384,  945, 1728,  2520, 3024,  3024, 2520,  1728,  945, 384;

  945, 3840, 4725, 11520, 9450, 16128, 9450, 11520, 4725, 3840, 945;

MAPLE

A172089 := proc(n, m)

        factorial(n)/doublefactorial(m)/doublefactorial(n-m) ;

end proc:

seq(seq(A172089(n, m), m=0..n), n=0..10) ; # R. J. Mathar, Oct 11 2011

MATHEMATICA

binomialn[n_, k_] = n!/(Factorial2[n-k]*Factorial2[k]); Table[binomialn[n, k], {n, 0, 10}, {k, 0, n}]//Flatten

PROG

(PARI)

f2(n) = prod(i=0, (n-1)\2, n - 2*i );

T(n, k) = n!/(f2(k)*f2(n-k));

for(n=0, 10, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Dec 05 2019

(MAGMA)

F2:=func< n | &*[n..2 by -2] >;

[Factorial(n)/(F2(k)*F2(n-k)): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 05 2019

(Sage)

def T(n, k): return factorial(n)/((k).multifactorial(2)*(n-k).multifactorial(2))

[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 05 2019

CROSSREFS

Cf. A000142, A006882.

Sequence in context: A039913 A108617 A092683 * A057475 A024376 A230128

Adjacent sequences:  A172086 A172087 A172088 * A172090 A172091 A172092

KEYWORD

nonn,tabl,easy

AUTHOR

Roger L. Bagula, Jan 25 2010

STATUS

approved

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Last modified July 16 11:09 EDT 2020. Contains 335784 sequences. (Running on oeis4.)