A172089 Triangle T(n,m) = n!/(m!!*(n-m)!!) read by rows, where (.)!! = A006882(.) are double factorials.

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

Offset: 0

Author

Roger L. Bagula, Jan 25 2010

Keywords

• nonn
• tabl
• easy

Comments

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

Examples

			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;
		

Links

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

Crossrefs

Cf. A000142, A006882.

Programs

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

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

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

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

Formula

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