A072019 -id:A072019 - cp's OEIS frontend

A091534 Generalized Stirling2 array (5,2).

1, 20, 10, 1, 1120, 1040, 290, 30, 1, 123200, 161920, 71320, 14040, 1340, 60, 1, 22422400, 37452800, 22097600, 6263040, 958720, 82800, 4000, 100, 1, 6098892800, 12222918400, 8928102400, 3257116800, 675281600, 84782880, 6625920, 322000

Offset: 1

Wolfdieter Lang, Jan 23 2004

The row length sequences for this array is [1,3,5,7,9,11,...]=A005408(n-1), n>=1.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
W. Lang, First 6 rows.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Cf. A078740 (3, 2)-Stirling2, A090438 (4, 2)-Stirling2.

Cf. A072019 (row sums), A091537 (alternating row sums).

a[n_, k_] := (-1)^k/k!*Sum[(-1)^p*Binomial[k, p]*Product[FactorialPower[p + 3*(j - 1), 2], {j, 1, n}], {p, 2, k}]; Table[a[n, k], {n, 1, 8}, {k, 2, 2 n}] // Flatten (* Jean-François Alcover, Sep 01 2016 *)

a(n, k)=(((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*product(fallfac(p+3*(j-1), 2), j=1..n), p=2..k), n>=1, 2<=k<=2*n, else 0. From eq. (12) of the Blasiak et al. reference with r=5, s=2.

Recursion: a(n, k)=sum(binomial(2, p)*fallfac(3*(n-1)+k-p, 2-p)*a(n-1, k-p), p=0..2), n>=2, 2<=k<=2*n, a(1, 2)=1, else 0. Rewritten from eq.(19) of the Schork reference with r=5, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle).

A072020 Sum of an infinite series: a(n) = Sum_{k = 0..infinity} ((1/27) * (3^n)^3 * Gamma(n+1/3k+1/3) Gamma(n+1/3k+2/3) Gamma(n+1/3k+1)) / (Gamma(4/3+1/3k) * Gamma(5/3+1/3k) Gamma(2+1/3k) exp(1) * k!).

Original entry on oeis.org

1, 229, 207775, 472630861, 2148321709561, 17028146983530961, 214877019857456672479, 4044349155369603186936985, 108105412214943249140163409201, 3949854849387058592656207156530781, 191308664212963089686669131219301608831

Offset: 1

Karol A. Penson, Jun 05 2002

nonn

			a(2) = 3!*LaguerreL(3, 3,-1) = 229, special value of associated Laguerre polynomial.

Sean A. Irvine, Table of n, a(n) for n = 1..25
Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.

Cf. A072019.

a[n_] := Sum[ 1/27*(3^n)^3 * Gamma[n + 1/3*k + 1/3] * Gamma[n + 1/3*k + 2/3] * Gamma[n + 1/3*k + 1] / Gamma[ 4/3 + 1/3*k] / Gamma[5/3 + 1/3*k] / Gamma[2 + 1/3*k] / Exp[1] / k!, {k, 0, Infinity}] (* Robert G. Wilson v, Jun 13 2002 *)

Representation as n-th moment of a positive function on a positive half-axis: a(n) = Integral_{x=0..oo} x^n*(exp(-x^(1/3))*BesselI(3, 2*x^(1/6))/(3*exp(1)*x^(7/6))) dx, n >= 1. This representation is unique.

a(9) from Robert G. Wilson v, Jun 13 2002

a(10) from Sean A. Irvine, Aug 26 2024

A091748 Generalized Bell numbers B_{6,2}.

Original entry on oeis.org

1, 43, 5083, 1160113, 432168721, 238012552651, 181520958432283, 182989529196234433, 235492729726705299073, 376560458072018837889931, 732162019709408940671604091, 1700645336651586566571229542193

Offset: 1

Wolfdieter Lang, Feb 27 2004

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Cf. A072019 ( B_{5, 2}).

a[n_] := Sum[Product[FactorialPower[k+4*(j-1), 2], {j, 1, n}]/k!, {k, 2, Infinity}]/E; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 12}] (* Jean-François Alcover, Sep 01 2016 *)

a(n)=sum(A091746(n, k), k=2..2*n)= sum((1/k!)*product(fallfac(k+4*(j-1), 2), j=1..n), k=2..infinity)/exp(1), n>=1. From eq.(9) of the Blasiak et al. reference with r=6, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle). a(0) := 1 may be added.

A091537 Alternating row sums of array A091534 (generalized Stirling2 array (5,2)).

Original entry on oeis.org

1, 11, 341, 19841, 1683981, 143771891, -15351301839, -27396364105599, -24059921739904039, -21285850978489377989, -20814945866103868804819, -22980130985849165090290239, -28698856680135265507625861339, -40335646598356375740067161474269

Offset: 1

Wolfdieter Lang, Jan 23 2004

Cf. A072019 (row sums of A091534).

a(n)=sum(((-1)^k)*A091534(n, k), k=2..2*n), n>=1.

A072598 a(n)= 3^(4n-4) Sum_{k>=0} ( Gamma(n+k/3+1/3) / Gamma(4/3+k/3) ) * (Gamma(n+k/3+2/3) / Gamma(5/3+k/3) ) * (Gamma(n+k/3+1) / Gamma(2+k/3) ) * Gamma(n+k/3+4/3) / ( Gamma(7/3+k/3) * k! *exp(1)).

Original entry on oeis.org

1, 1961, 22982765, 897960515649, 87104111341922641, 17553971396873140572281, 6522485663882405640795371581, 4101670732571144797304609148820545, 4092893743093429344164150395340192148609, 6162975970715988703282664052430391759867866441

Offset: 1

Karol A. Penson, Jun 23 2002

nonn

Gamma in the definition is the standard Capital-Greek-Gamma function.

Cf. A072019, A072020.

More terms from Sean A. Irvine, Oct 13 2024

cp's OEIS Frontend

A091534 Generalized Stirling2 array (5,2).

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A072020 Sum of an infinite series: a(n) = Sum_{k = 0..infinity} ((1/27) * (3^n)^3 * Gamma(n+1/3*k+1/3) * Gamma(n+1/3*k+2/3) * Gamma(n+1/3*k+1)) / (Gamma(4/3+1/3*k) * Gamma(5/3+1/3*k) * Gamma(2+1/3*k) * exp(1) * k!).

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A091748 Generalized Bell numbers B_{6,2}.

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A091537 Alternating row sums of array A091534 (generalized Stirling2 array (5,2)).

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A072598 a(n)= 3^(4*n-4)* Sum_{k>=0} ( Gamma(n+k/3+1/3) / Gamma(4/3+k/3) ) * (Gamma(n+k/3+2/3) / Gamma(5/3+k/3) ) * (Gamma(n+k/3+1) / Gamma(2+k/3) ) * Gamma(n+k/3+4/3) / ( Gamma(7/3+k/3) * k! *exp(1)).

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A072020 Sum of an infinite series: a(n) = Sum_{k = 0..infinity} ((1/27) * (3^n)^3 * Gamma(n+1/3k+1/3) Gamma(n+1/3k+2/3) Gamma(n+1/3k+1)) / (Gamma(4/3+1/3k) * Gamma(5/3+1/3k) Gamma(2+1/3k) exp(1) * k!).

A072598 a(n)= 3^(4n-4) Sum_{k>=0} ( Gamma(n+k/3+1/3) / Gamma(4/3+k/3) ) * (Gamma(n+k/3+2/3) / Gamma(5/3+k/3) ) * (Gamma(n+k/3+1) / Gamma(2+k/3) ) * Gamma(n+k/3+4/3) / ( Gamma(7/3+k/3) * k! *exp(1)).