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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).
2
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).
LINKS
B. T. Gill, Problem 1729Math. Magazine, vol. 79, No. 4, 2006, p. 313, problem 1729.
FORMULA
T(n,0) = 2^n * n! = A000165(n).
T(n,n) = (-1)^n*A001147(n).
From Peter Bala, Aug 09 2024: (Start)
The polynomial P(n, x) = Sum_{k = 0..n} T(n, k)*x^(n-k) satisfies the functional equation P(n, 1 - x) = (-1)^n*P(n, x).
P(n, x) = (2*n - 1)*(2*x - 1)*P(n-1, x) + 4*(n - 1)^2*x*(1 - x)*P(n-2, x) with P(0, x) = 1 and P(1, x) = 2*x - 1.
P(n, 1/2) = A177145(n+1); P(n, -1/2) = (-1)^n*A331817(n).
Conjecture 1: for n >= 1, the zeros of P(n, x) lie on the vertical line Re(x) = 1/2 in the complex plane; that is, the family of polynomials {P(n, x) : n >= 1} satisfies a Riemann hypothesis.
Set u = x^2 and define p(n, u) = P(n, 1/2 + x) if n is even, else p(n, x) = (1/x)* P(n, 1/2 + x). The first few polynomials are p(0, u) = 1, p(1, u) = 2, p(2, u) = 8*u + 1, p(3, u) = 48*u + 18 and p(4, u) = 384*u^2 + 288*u + 9.
Conjecture 2: for n >= 2, the zeros of p(n+1, u) are negative and interlace the zeros of p(n, u). (End)
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
KEYWORD
sign,tabl
AUTHOR
Emeric Deutsch, Oct 14 2006
STATUS
approved