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

0,5

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