A091541 Four times triple factorials (3*n-2)!!! with leading 1 added.

1, 4, 4, 16, 112, 1120, 14560, 232960, 4426240, 97377280, 2434432000, 68164096000, 2113086976000, 71844957184000, 2658263415808000, 106330536632320000, 4572213075189760000, 210321801458728960000, 10305768271477719040000, 535899950116841390080000, 29474497256426276454400000

Offset: 0

Author

Wolfdieter Lang, Feb 13 2004

Keywords

• nonn
• easy
• changed

Comments

The exponential (or binomial) convolution of a(n) with A051606(n) gives A091540.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..381

Crossrefs

Cf. A007559, A051606, A091540.

Programs

Magma

m:=50; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(3 - 2*(1 - 3*x)^(2/3))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

Mathematica

With[{nmax = 50}, CoefficientList[Series[3 - 2*(1 - 3*x)^(2/3), {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Aug 15 2018 *)

PARI

my(x='x+O('x^50)); Vec(serlaplace(3 - 2*(1 - 3*x)^(2/3))) \\ G. C. Greubel, Aug 15 2018

Formula

a(0) = 1, a(n) = 4*(3*n-2)!!! = 4*A007559(n-1), n>=1.

E.g.f. 3-2*(1-3*x)^(2/3).

E.g.f. for a(n+1)/4 = A007559(n), n>=0: (1-3*x)^(-1/3).

G.f.: 3-G(0), where G(k)= 1 + 1/(1 - x*(3*k-2)/(x*(3*k-2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 11 2013

From Amiram Eldar, Aug 30 2025: (Start)

a(n) ~ 4 * sqrt(2*Pi) * 3^(n-1) * n^(n-7/6) / (Gamma(1/3) * exp(n)).

Sum_{n>=0} 1/a(n) = (5 + (e/9)^(1/3) * (Gamma(1/3) - Gamma(1/3, 1/3))) / 4. (End)