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
Examples
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, ...
Programs
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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);
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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 *) -
Sage
from sage.functions.hypergeometric import closed_form def A246256_row(n): R.
= 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)
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