A123516 Triangle read by rows: T(n,k) = (-1)^k * n! * 2^(n-2*k) * binomial(n,k) * binomial(2*k,k) (0<=k<=n).

1, 2, -1, 8, -8, 3, 48, -72, 54, -15, 384, -768, 864, -480, 105, 3840, -9600, 14400, -12000, 5250, -945, 46080, -138240, 259200, -288000, 189000, -68040, 10395, 645120, -2257920, 5080320, -7056000, 6174000, -3333960, 1018710, -135135, 10321920, -41287680, 108380160, -180633600, 197568000

Offset

0,2

Comments

Row sums yield the double factorial numbers (A001147).

References

B. T. Gill, Math. Magazine, vol. 79, No. 4, 2006, p. 313, problem 1729.

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Formula

T(n,0) = 2^n * n! = A000165(n).

T(n,n) = (-1)^n*A001147(n).

Example

Triangle begins:
1;
2,     -1;
8,     -8,      3;
48,    -72,     54,     -15;
384,   -768,    864,    -480,    105;
3840,  -9600,   14400,  -12000,  5250,   -945;
46080, -138240, 259200, -288000, 189000, -68040, 10395;
...

Maple

T:=(n, k)->(-1)^k*n!*2^(n-2*k)*binomial(n, k)*binomial(2*k, k): for n from 0 to 8 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form

Mathematica

Table[(-1)^k*n! 2^(n - 2 k)*Binomial[n, k]*Binomial[2*k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Oct 14 2017 *)

Prog

(PARI) for(n=0, 10, for(k=0, n, print1((-1)^k*n!*2^(n-2*k)*binomial(n, k)* binomial(2*k, k), ", "))) \\ G. C. Greubel, Oct 14 2017
(Magma) /* As triangle * / [[(-1)^k*Factorial(n)*2^(n-2*k)* Binomial(n, k)*Binomial(2*k, k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Oct 15 2017

Crossrefs

Cf. A001147, A000165.

Sequence in context: A145901 A321369 A286724 * A193604 A016446 A254794

Adjacent sequences: A123513 A123514 A123515 * A123517 A123518 A123519

Keyword

sign,tabl

Author

Emeric Deutsch, Oct 14 2006

Status

approved