A091539 -id:A091539 - cp's OEIS frontend

A091540 Rescaled second column A091539 of array A091534 ((5,2)-Stirling2).

1, 13, 184, 3040, 58360, 1283800, 31917760, 886123840, 27192323200, 914387689600, 33446228569600, 1322364153510400, 56203860301388800, 2555756347720576000, 123819357959385088000, 6367367706293321728000

Offset: 2

Wolfdieter Lang, Feb 13 2004

A certain difference of two triple factorial sequences.

If offset 0: exponential (also called binomial) convolution of A091541 and A051606.

G. C. Greubel, Table of n, a(n) for n = 2..380

Cf. A091541.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (3-2*(1-3*x)^(2/3))/(1-3*x)^3 )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

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

x='x+O('x^30); Vec(serlaplace((1 - 2*x - (1 - 3*x)^(2/3))/(2*(1 - 3*x)))) \\ G. C. Greubel, Aug 15 2018

a(n)= (5*2/fac3(3*n-1))*A091539(n), n>=2, with fac3(3*n-1) := A008544(n) (triple factorials).
E.g.f.: (1-2*x-(1-3*x)^(2/3))/(2*(1-3*x))= (1/2-x+int((1-3*x)^(-1/3), x))/(1-3*x).
E.g.f. with offset 0: (3-2*(1-3*x)^(2/3))/(1-3*x)^3.
a(n)=(fac3(3*n) - 3*fac3(3*n-2))/3! with fac3(3*n) := A032031(n)= n!*3^n and fac3(3*n-2) := A007559(n).
a(n) ~ 3^(n-1) * n! / 2. - Vaclav Kotesovec, Aug 16 2018

A091550 Second column (k=3) sequence of array A091746 ((6,2)-Stirling2) divided by 12.

Original entry on oeis.org

1, 160, 39900, 15120000, 8202070800, 6058891238400, 5860547004312000, 7196668193594880000, 10944624305020966560000, 20199809308312018344960000, 44490168120726255724917120000, 115290834599202214240544256000000, 347284815748143369922163257920000000

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. A091539 (second column of (5, 2)-Stirling2 array), A091550 (second column of (7, 2)-Stirling2 array), A091746.

a[n_] := 2^(4*n) * Pochhammer[1/2, n] * (-3 * Pochhammer[1/4, n] + Pochhammer[3/4, n])/(3!*12); Array[a, 20, 2] (* Amiram Eldar, Aug 30 2025 *)

a(n) = (2^(4*n)) * risefac(1/2, n) * (-3*risefac(1/4, n) + risefac(3/4, n))/(3!*12), n>=2, with risefac(x, n) = Pochhammer(x, n).
E.g.f.: (hypergeom([1/2, 3/4], [], 16*x) - 3*hypergeom([1/4, 1/2], [], 16*x) + 2)/(3!*12).
a(n) = (2^n) * Product_{j=0..n-1} (2*j+1) * (-3*Product_{j=0..n-1} (4*j+1) + Product_{j=0..n-1} (4*j+3))/(3!*12), n>=2. From eq.12 of the Blasiak et al. reference with r=6, s=2, k=3.
a(n) ~ sqrt(Pi) * 2^(4*n-2) * n^(2*n+1/4) / (9 * Gamma(3/4) * exp(2*n)). - Amiram Eldar, Aug 30 2025

A091740 Third column (k=4) sequence of array A091534 ((5,2)-Stirling2).

Original entry on oeis.org

1, 290, 71320, 22097600, 8928102400, 4644244774400, 3046988353024000, 2470747704449024000, 2431736840968314880000, 2859398101389251502080000, 3962371103307529193881600000, 6394280010754055221811609600000, 11892513203530676764397417267200000, 25260371493666997186451230294016000000

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. A091539 (second column of array A091534 divided by 10), A007661.

a[n_] := 3^(2*n) * (6 * Pochhammer[2/3, n] * Pochhammer[1/3, n] - 4 * n! * Pochhammer[2/3, n] + n! * Pochhammer[4/3, n])/4!; Array[a, 20, 2] (* Amiram Eldar, Aug 30 2025 *)

a(n) = A091534(n, 4), n>=2.
a(n) = (3^(2*n)) * (6*risefac(2/3, n) * risefac(1/3, n) - 4*n!*risefac(2/3, n) + risefac(4/3, n)*n!)/4!, with risefac(x, n) = Pochhammer(x, n).
E.g.f.: (6*hypergeom([2/3, 1/3], [], 9*x) - 4*hypergeom([1, 2/3], [], 9*x) + hypergeom([4/3, 1], [], 9*x) - 3)/4!.
a(n) = (6*fac3(3*n-2)*fac3(3*n-1)-4*fac3(3*n-1)*fac3(3*n)+fac3(3*n)*fac3(3*n+1))/4!, n>=2, with fac3(n) = A007661(n) (triple factorials). Rewritten from eq.12 of the Blasiak et al. reference for r=5, s=2, k=4.

cp's OEIS Frontend

A091540 Rescaled second column A091539 of array A091534 ((5,2)-Stirling2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A091550 Second column (k=3) sequence of array A091746 ((6,2)-Stirling2) divided by 12.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A091740 Third column (k=4) sequence of array A091534 ((5,2)-Stirling2).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula