A091540 -id:A091540 - cp's OEIS frontend

A091539 Second column (k=3) of array A091534 ((5,2)-Stirling2) divided by 10.

1, 104, 16192, 3745280, 1222291840, 537758144000, 307503360102400, 221965373351321600, 197530935371241472000, 212553938009841139712000, 272115940122123843665920000, 408828811133790954169303040000, 712427095375430807967713198080000, 1425431682224708301179257251430400000

Offset: 2

Wolfdieter Lang, Feb 13 2004

Pawel Blasiak, Karol A. Penson, and Allan I. Solomon, The general boson normal ordering problem, Physics Letters A, Vol. 309, No. 3-4 (2003), pp. 198-205; arXiv preprint, arXiv:quant-ph/0402027, 2004.

Cf. A091534, A091540.

Cf. A032031, A008544, A007559, A007661.

a[n_] := 3^(2*n) * Pochhammer[2/3, n] * (n! - 3 * Pochhammer[1/3, n])/(3!*10); Array[a, 20, 2] (* Amiram Eldar, Aug 30 2025 *)

a(n) = A091534(n, 3)/10, n >= 2.
a(n) = Product_{j=0..n-1} (3*j + 2)*(Product_{j=0..n-1} (3*(j+1)) - 3*Product_{j=0..n-1} (3*j + 1))/(3!*10). From eq. (12) of the Blasiak et al. reference for r=5, s=2 and k=3.
a(n) = (3^(2*n))*risefac(2/3, n)*(n!-3*risefac(1/3, n))/(3!*10), with risefac(x, n) = Pochhammer(x, n).
a(n) = (fac3(3*n-1)/10)*(fac3(3*n) - 3*fac3(3*n-2))/3!, with fac3(3*n) = A032031(n) = n!*3^n, fac3(3*n-1) = A008544(n) and fac3(3*n-2) = A007559(n) (triple factorials: fac3(n) = A007661(n)).
E.g.f.: (hypergeom([2/3, 1], [], 9*x)-3*hypergeom([1/3, 2/3], [], 9*x)+2)/(3!*10).
a(n) ~ Pi * 3^(2*n) *  n^(2*n + 2/3) / (30 * Gamma(2/3) * exp(2*n)). - Amiram Eldar, Aug 30 2025

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

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Feb 13 2004

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

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

Cf. A007559, A051606, A091540.

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

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 *)

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

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)

cp's OEIS Frontend

A091539 Second column (k=3) of array A091534 ((5,2)-Stirling2) divided by 10.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula