A246256 Triangular array read by rows. Row n lists the coefficients of the closed form of hypergeometric([1/2, -n/2, (1-n)/2], [], 4*z).

1, 1, 1, 1, 3, 1, 9, 6, 1, 45, 10, 1, 225, 135, 15, 1, 1575, 315, 21, 1, 11025, 6300, 630, 28, 1, 99225, 18900, 1134, 36, 1, 893025, 496125, 47250, 1890, 45, 1, 9823275, 1819125, 103950, 2970, 55, 1, 108056025, 58939650, 5457375, 207900, 4455, 66, 1

Offset

0,5

Links

Table of n, a(n) for n=0..48.

Formula

For the e.g.f. and a recurrence see the Maple program.

T(n, k) = (Product_{j=0..(floor(n/2) - k)} (2*j - 1)^2)*binomial(n, 2*k + (n mod 2)). - Detlef Meya, May 05 2024

Example

Triangle starts:
[ 0] 1,
[ 1] 1,
[ 2] 1, 1,
[ 3] 3, 1,
[ 4] 9, 6, 1,
[ 5] 45, 10, 1,
[ 6] 225, 135, 15, 1,
[ 7] 1575, 315, 21, 1,
[ 8] 11025, 6300, 630, 28, 1,
[ 9] 99225, 18900, 1134, 36, 1,
[10] 893025, 496125, 47250, 1890, 45, 1,
[11] 9823275, 1819125, 103950, 2970, 55, 1,
...

Maple

g := exp(x*z)/sqrt((1-z)/(1+z)); gser := n -> series(g, z, n+2):
seq(seq(coeff(n!*coeff(gser(n), z, n), x, 2*i+irem(n, 2)), i=0..iquo(n, 2)), n=0..12);
# Recurrence for A138022 from Robert Israel.
T := proc(n, k) option remember;
if k < 0 or n < k then 0 elif k = n then 1 elif k = n-1 then n
elif k = 0 then T(n-1, k)+(n-2)*(n-1)*T(n-2, k)
else T(n-1, k)+T(n-1, k-1)+(n-2)*(n-1)*(T(n-2, k)-T(n-3, k-1)) fi end:
A246256_row := n -> seq(T(n, 2*k+(n mod 2)), k=0..iquo(n, 2)):
seq(A246256_row(n), n=0..12);

Mathematica

row[n_] := HypergeometricPFQ[{1/2, -n/2, (1-n)/2}, {}, 4z] // FunctionExpand // CoefficientList[#, z]& // Reverse;
Table[row[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Aug 02 2019 *)
T[n_, k_] := Product[(2*j - 1)^2, {j, 0, Floor[n/2] - k}]*Binomial[n, 2*k + Mod[n, 2]]; Flatten[Table[T[n, k], {n, 0, 12}, {k, 0 , Floor[n/2]}]] (*  Detlef Meya, May 05 2024 *)

Prog

(Sage)
from sage.functions.hypergeometric import closed_form
def A246256_row(n):
    R.<z> = ZZ[]
    h = hypergeometric([1/2, -n/2, (1-n)/2], [], 4*z)
    T = R(closed_form(h)).coefficients()
    return T[::-1]
for n in range(13): A246256_row(n)

Crossrefs

Cf. A081919 (row sums), A138022, A246257.

Sequence in context: A164942 A236420 A187537 * A157393 A242402 A217629

Adjacent sequences: A246253 A246254 A246255 * A246257 A246258 A246259

Keyword

tabf,nonn

Author

Peter Luschny, Aug 21 2014

Status

approved