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A172088 Triangle: T(n,m) = n!! - m!! - (n-m)!! read by rows 0 <= m <= n, where ()!! are the double factorials. 1
-1, -1, -1, -1, 0, -1, -1, 0, 0, -1, -1, 4, 4, 4, -1, -1, 6, 10, 10, 6, -1, -1, 32, 38, 42, 38, 32, -1, -1, 56, 88, 94, 94, 88, 56, -1, -1, 278, 334, 366, 368, 366, 334, 278, -1, -1, 560, 838, 894, 922, 922, 894, 838, 560, -1, -1, 2894, 3454, 3732, 3784, 3810, 3784, 3732, 3454, 2894, -1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,12

COMMENTS

Row sums are {-1, -2, -2, -2, 10, 30, 180, 474, 2322, 6426, 31536, ...}; n-th row sum is (n+1)*n!! - 2*A129981(n).

LINKS

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

FORMULA

T(n,m) = A006882(n) - A006882(m) - A006882(n-m).

EXAMPLE

Triangle begins

  -1;

  -1,   -1;

  -1,    0,   -1;

  -1,    0,    0,   -1;

  -1,    4,    4,    4,   -1;

  -1,    6,   10,   10,    6,   -1;

  -1,   32,   38,   42,   38,   32,   -1;

  -1,   56,   88,   94,   94,   88,   56,   -1;

  -1,  278,  334,  366,  368,  366,  334,  278,   -1;

  -1,  560,  838,  894,  922,  922,  894,  838,  560,   -1;

  -1, 2894, 3454, 3732, 3784, 3810, 3784, 3732, 3454, 2894, -1;

MAPLE

A172088 := proc(n, m)

        doublefactorial(n)-doublefactorial(m)-doublefactorial(n-m) ;

end proc:

seq(seq(A172088(n, m), m=0..n), n=0..10) ; # R. J. Mathar, Oct 11 2011

MATHEMATICA

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

PROG

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

T(n, k) = f2(n) - f2(k) - f2(n-k); \\ G. C. Greubel, Dec 05 2019

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

[F2(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 (n).multifactorial(2) - (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. A006882, A129981.

Sequence in context: A243756 A172985 A117683 * A159891 A201941 A180060

Adjacent sequences:  A172085 A172086 A172087 * A172089 A172090 A172091

KEYWORD

sign,tabl,easy

AUTHOR

Roger L. Bagula, Jan 25 2010

STATUS

approved

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Last modified July 14 19:48 EDT 2020. Contains 335729 sequences. (Running on oeis4.)