A395577 1 = Sum_{n>=0} a(n) * x^n * (1 - (4*n+1)*x)^2.

1, 2, 19, 292, 6053, 156454, 4821751, 172091336, 6967703113, 315139591882, 15732501109211, 858638989668844, 50831174705612269, 3242711167070593262, 221681733241307827903, 16162941921207795907984, 1251634963086341967917201, 102570238312410247255627666

Offset

0,2

Links

Table of n, a(n) for n=0..17.

Formula

a(n) = 2*(4*n - 3)*a(n-1) - (4*n - 7)^2*a(n-2).

a(n) ~ c * 2^(2*n) * exp(2*sqrt(n) - n) * n^(n - 1/2) * (1 - 5/(48*sqrt(n))), where c = 0.1931408318743114767472954651005984550581402157927430919...

Mathematica

RecurrenceTable[{(-7+4*n)^2*a[n-2] - 2*(-3+4*n)*a[n-1] + a[n] == 0, a[1]==2, a[2]==19}, a, {n, 0, 20}]
(* Alternative: *)
nmax = 20; Round[Assuming[{x > 0}, CoefficientList[Series[E^(4*x/(1 - 4*x)) * ((2*(2*HypergeometricU[-1/4, 1, -1] + HypergeometricU[3/4, 1, -1]) * LaguerreL[-3/4, 1/(-1 + 4*x)] + HypergeometricU[3/4, 1, 1/(-1 + 4*x)] * (-2*LaguerreL[-3/4, -1] + LaguerreL[1/4, -1])) / ((1 - 4*x)^(1/4) * (4*HypergeometricU[-1/4, 1, -1] * LaguerreL[-3/4, -1] + HypergeometricU[3/4, 1, -1] * LaguerreL[1/4, -1]))), {x, 0, nmax}], x]] * Range[0, nmax]!] (* Vaclav Kotesovec, May 09 2026 *)

Prog

(PARI) {a(n)=polcoeff(1-sum(m=0, n-1, a(m)*x^m*(1-(4*m+1)*x+x*O(x^n))^2), n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A000142, A002720, A219779, A219780.

Cf. A001147, A222080, A395559.

Cf. A007559, A395560, A395576.

Sequence in context: A379287 A252710 A065923 * A293946 A365652 A094476

Adjacent sequences: A395574 A395575 A395576 * A395578 A395579 A395580

Keyword

nonn,changed

Author

Vaclav Kotesovec, Apr 29 2026

Status

approved