A395930 Triangle read by rows: T(n, k) = 4^n * [x^k] hypergeom([-n, -n, 1/2], [1], x).

1, 4, 2, 16, 32, 12, 64, 288, 432, 120, 256, 2048, 6912, 7680, 1680, 1024, 12800, 76800, 192000, 168000, 30240, 4096, 73728, 691200, 3072000, 6048000, 4354560, 665280, 16384, 401408, 5419008, 37632000, 131712000, 213373440, 130394880, 17297280

Offset

0,2

Links

Table of n, a(n) for n=0..35.

Formula

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.

Example

Triangle starts:
  [0]     1;
  [1]     4,     2;
  [2]    16,    32,     12;
  [3]    64,   288,    432,     120;
  [4]   256,  2048,   6912,    7680,    1680;
  [5]  1024, 12800,  76800,  192000,  168000,   30240;
  [6]  4096, 73728, 691200, 3072000, 6048000, 4354560, 665280;

Maple

p := (n, x) -> 4^n * hypergeom([-n, -n, 1/2], [1], x):
T := (n, k) -> coeff(simplify(p(n, x)), x, k):
for n from 0 to 6 do seq(T(n, k), k = 0..n) od;

Prog

(SageMath)
def P(n):
    x = SR.var('x')
    poly = c = SR(1)
    for k in range(n):
        c = (c * (k - n)^2 * (2*k + 1) / 2) / (k + 1)^2
        poly += c * x^(k + 1)
    return 4^n * poly
for n in range(7): print(n, P(n).list())
(PARI) T(n, k) = 4^n * polcoef(hypergeom([-n, -n, 1/2], [1], x), k); \\ Michel Marcus, May 12 2026

Crossrefs

Cf. A393915, A001813 (main diagonal).

Sequence in context: A264195 A182872 A137393 * A356569 A122749 A390029

Adjacent sequences: A395927 A395928 A395929 * A395931 A395932 A395933

Keyword

nonn,tabl

Author

Peter Luschny, May 12 2026

Status

approved