A395930 - OEIS

%I #18 May 12 2026 12:27:00

%S 1,4,2,16,32,12,64,288,432,120,256,2048,6912,7680,1680,1024,12800,

%T 76800,192000,168000,30240,4096,73728,691200,3072000,6048000,4354560,

%U 665280,16384,401408,5419008,37632000,131712000,213373440,130394880,17297280

%N Triangle read by rows: T(n, k) = 4^n * [x^k] hypergeom([-n, -n, 1/2], [1], x).

%F T(n, k) = [x^k] Sum_{k=0..n} 4^(n-k)*binomial(n, k)^2*binomial(2*k, k)*k!*x^k, see A393915.

%e Triangle starts:

%e   [0]     1;

%e   [1]     4,     2;

%e   [2]    16,    32,     12;

%e   [3]    64,   288,    432,     120;

%e   [4]   256,  2048,   6912,    7680,    1680;

%e   [5]  1024, 12800,  76800,  192000,  168000,   30240;

%e   [6]  4096, 73728, 691200, 3072000, 6048000, 4354560, 665280;

%p p := (n, x) -> 4^n * hypergeom([-n, -n, 1/2], [1], x):

%p T := (n, k) -> coeff(simplify(p(n, x)), x, k):

%p for n from 0 to 6 do seq(T(n, k), k = 0..n) od;

%o (SageMath)

%o def P(n):

%o     x = SR.var('x')

%o     poly = c = SR(1)

%o     for k in range(n):

%o         c = (c * (k - n)^2 * (2*k + 1) / 2) / (k + 1)^2

%o         poly += c * x^(k + 1)

%o     return 4^n * poly

%o for n in range(7): print(n, P(n).list())

%o (PARI) T(n, k) = 4^n * polcoef(hypergeom([-n, -n, 1/2], [1], x), k); \\ _Michel Marcus_, May 12 2026

%Y Cf. A393915, A001813 (main diagonal).

%K nonn,tabl

%O 0,2

%A _Peter Luschny_, May 12 2026