A395560 1 = Sum_{n>=0} a(n) * x^n * (1 - (3*n+1)*x)^2.

1, 2, 15, 178, 2825, 55650, 1303375, 35281850, 1081883025, 37017735850, 1396816316975, 57580706746050, 2573147578118425, 123849623782329650, 6385330868142248175, 350979056608505903050, 20483596432787526205025, 1264720766629579077238650, 82349844694353373614554575

Offset

0,2

Links

Table of n, a(n) for n=0..18.

Formula

a(n) = 2*(3*n - 2)*a(n-1) - (3*n - 5)^2*a(n-2).

a(n) ~ c * 3^n * exp(2*sqrt(n) - n) * n^(n - 5/12) * (1 - 1/(48*sqrt(n))), where c = 0.24572580484444376099872807858037743671469557301534443383198986...

Mathematica

RecurrenceTable[{(3*n-5)^2*a[n-2] - 2*(3*n-2)*a[n-1] + a[n] == 0, a[1] == 2, a[2] == 15}, a, {n, 0, 20}]
(* or *)
nmax = 20; Round[Assuming[{x > 0}, CoefficientList[Series[E^(3*x/(1 - 3*x)) * (((3*HypergeometricU[-1/3, 1, -1] + 2*HypergeometricU[2/3, 1, -1]) * LaguerreL[-2/3, 1/(-1 + 3*x)] + HypergeometricU[2/3, 1, 1/(-1 + 3*x)] * (-2*LaguerreL[-2/3, -1] + LaguerreL[1/3, -1])) / ((1 - 3*x)^(1/3) * (3*HypergeometricU[-1/3, 1, -1] * LaguerreL[-2/3, -1] + HypergeometricU[2/3, 1, -1] * LaguerreL[1/3, -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-(3*m+1)*x+x*O(x^n))^2), n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A222080, A219779, A395559.

Sequence in context: A215368 A200795 A121427 * A364340 A099709 A208409

Adjacent sequences: A395557 A395558 A395559 * A395561 A395562 A395563

Keyword

nonn

Author

Vaclav Kotesovec, Apr 28 2026

Status

approved