This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A123516 #28 Aug 16 2024 06:01:28
%S A123516 1,2,-1,8,-8,3,48,-72,54,-15,384,-768,864,-480,105,3840,-9600,14400,
%T A123516 -12000,5250,-945,46080,-138240,259200,-288000,189000,-68040,10395,
%U A123516 645120,-2257920,5080320,-7056000,6174000,-3333960,1018710,-135135,10321920,-41287680,108380160,-180633600,197568000
%N 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).
%C A123516 Row sums yield the double factorial numbers (A001147).
%H A123516 G. C. Greubel, <a href="/A123516/b123516.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A123516 B. T. Gill, <a href="https://www.jstor.org/stable/27642959">Problem 1729</a>Math. Magazine, vol. 79, No. 4, 2006, p. 313, problem 1729.
%F A123516 T(n,0) = 2^n * n! = A000165(n).
%F A123516 T(n,n) = (-1)^n*A001147(n).
%F A123516 From _Peter Bala_, Aug 09 2024: (Start)
%F A123516 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).
%F A123516 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.
%F A123516 P(n, 1/2) = A177145(n+1); P(n, -1/2) = (-1)^n*A331817(n).
%F A123516 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.
%F A123516 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.
%F A123516 Conjecture 2: for n >= 2, the zeros of p(n+1, u) are negative and interlace the zeros of p(n, u). (End)
%e A123516 Triangle begins:
%e A123516 1;
%e A123516 2, -1;
%e A123516 8, -8, 3;
%e A123516 48, -72, 54, -15;
%e A123516 384, -768, 864, -480, 105;
%e A123516 3840, -9600, 14400, -12000, 5250, -945;
%e A123516 46080, -138240, 259200, -288000, 189000, -68040, 10395;
%e A123516 ...
%p A123516 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
%t A123516 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 *)
%o A123516 (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
%o A123516 (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
%Y A123516 Cf. A001147, A000165, A177145, A331817.
%K A123516 sign,tabl
%O A123516 0,2
%A A123516 _Emeric Deutsch_, Oct 14 2006