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URL: https://oeis.org/A302770

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A302770
a(n) = (4*n-2)*((n-1)*a(n-1) + ((n-2)!)^2), with a(1) = 0, n > 1.
1
0, 6, 130, 5516, 397800, 43770672, 6828599232, 1434021390720, 390054681930240, 133398762996833280, 56027485989309542400, 28349908516327342694400, 17009945189463951728640000, 11940981535393590884843520000, 9696077008988591505023631360000
OFFSET
1,2
FORMULA
a(n) = f3(n)*((n-1)!)^2, where f3(n) corresponds to the z values such that Sum_{k>=0} 1/(binomial(2*k,k)*(k+n)) = x*Pi*sqrt(3) - y*Pi^2 - z. (See examples for connection with a(n) in terms of material at Links section.)
f2(n) corresponds to the y values, so f2(n) = (1/9)*((2*n-1)!/((n-1)!)^2).
a(n) ~ Pi^(5/2) * 2^(2*n) * n^(2*n - 1/2) / (9*exp(2*n)). - Vaclav Kotesovec, Jan 19 2026
EXAMPLE
Examples ((3.79) - (3.83)) at page 14 in Links section as follows, respectively.
For n=1, f3(1) = 0, so a(1) = 0.
For n=2, f3(2) = 6, so a(2) = 6.
For n=3, f3(3) = 65/2, so a(3) = 130.
For n=4, f3(4) = 1379/9, so a(4) = 5516.
For n=5, f3(5) = 5525/8, so a(5) = 397800.
MATHEMATICA
RecurrenceTable[{a[n] == (4*n-2) * ((n-1)*a[n-1] + (n-2)!^2), a[1] == 0}, a, {n, 1, 15}] (* Vaclav Kotesovec, Apr 13 2018 *)
Table[FullSimplify[Pi^2*Gamma[2*n]/9 - Gamma[n]^2 * HypergeometricPFQ[{1, n, n}, {n + 1/2, n + 1}, 1/4]/n], {n, 1, 15}] (* Vaclav Kotesovec, Apr 13 2018 *)
nmax = 15; Table[CoefficientList[Expand[FunctionExpand[ Table[-Sum[1/(Binomial[2*j, j]*(j + m)), {j, 0, Infinity}], {m, 1, nmax}]]], Pi][[n, 1]]*(n - 1)!^2, {n, 1, nmax}] (* Vaclav Kotesovec, Apr 13 2018 *)
CROSSREFS
Cf. A302769.
Sequence in context: A000907 A188718 A077031 * A337971 A281402 A137038
KEYWORD
nonn
AUTHOR
Detlef Meya, Apr 13 2018
EXTENSIONS
More terms from Altug Alkan, Apr 13 2018
STATUS
approved