A091534 -id:A091534 - cp's OEIS frontend

A091535 First column (k=2) of array A091534 ((5,2)-Stirling2).

1, 20, 1120, 123200, 22422400, 6098892800, 2317579264000, 1172695107584000, 762251819929600000, 618948477782835200000, 613996889960572518400000, 730656299053081296896000000, 1027302756468632303435776000000, 1684776520608556977634672640000000

Offset: 1

Wolfdieter Lang, Jan 23 2004

The scaled sequence (2/(3n-1)!!!)*a(n) = (3*n-2)!!! = A007559(n), n>=1.

Third column of array A091752 ((-1, 2)-Stirling2).

Cf. A007559, A032031, A091534.

a := n -> 9^n*GAMMA(n+1/3)*GAMMA(n+2/3)*sqrt(3)/(4*Pi);
seq(a(n), n=1..16); # Peter Luschny, Sep 17 2014

a[n_] := (3*n-1)!/(2!*3^(n-1)*(n-1)!); Array[a, 15] (* Amiram Eldar, Sep 01 2025 *)

a(n) = (3*n-1)!/(2!*3^(n-(2-1))*(n-1)!) = ((3*n-1)!/2)/A032031(n-1).
a(n) = A091534(n, 2), n>=1.
E.g.f.: (hypergeom([1/3, 2/3, 1], [], 9*x)-1)/2.
a(n) = 9^n*Gamma(n+1/3)*Gamma(n+2/3)*sqrt(3)/(4*Pi). - Peter Luschny, Sep 17 2014
a(n) ~ (sqrt(3)/2) * (3*n/e)^(2*n). - Amiram Eldar, Sep 01 2025

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

Original entry on oeis.org

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

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

Original entry on oeis.org

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

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.

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.

A091746 Generalized Stirling2 array (6,2).

Original entry on oeis.org

1, 30, 12, 1, 2700, 1920, 426, 36, 1, 491400, 478800, 162540, 25344, 1956, 72, 1, 150368400, 181440000, 80451000, 17624880, 2130660, 147840, 5820, 120, 1, 69470200800, 98424849600, 52905560400, 14618016000, 2346624000, 232202880

Offset: 1

Wolfdieter Lang, Feb 27 2004

The sequence of row lengths 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.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

W. Lang, First 6 rows.

Cf. A078740 (3, 2)-Stirling2, A090438 (4, 2)-Stirling2, A091534 (5, 2)-Stirling2.
Cf. A091544 (first column), A091550 (second column divided by 12).
Cf. A091748 (row sums), A091750 (alternating row sums).

a[n_, k_] := (-1)^k/k! Sum[(-1)^p Binomial[k, p] Product[FactorialPower[p + 4*(j-1), 2], {j, 1, n}], {p, 2, k}]; Table[a[n, k], {n, 1, 8}, {k, 2, 2n} ] // Flatten (* Jean-François Alcover, Sep 01 2016 *)

a(n, k)=(((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*product(fallfac(p+4*(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=6, s=2.

Recursion: a(n, k)=sum(binomial(2, p)*fallfac(4*(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=6, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle).

A072019 Generalized Bell numbers B_{5,2}.

Original entry on oeis.org

1, 31, 2481, 371881, 89281461, 31274052351, 15020526041221, 9461707887414161, 7560380738419084201, 7466459670646734124671, 8925493084998518977531001, 12696331763378714706289411961, 21186586117648690791837844061341, 40976310644118022811682503135528671, 90905327647146969025291153908894514381, 229256189615621846477632508681520371943201

Offset: 1

Karol A. Penson, Jun 05 2002

nonn

a(n), n=1,2... can be calculated as n-th moment of a positive function on a positive half-axis. This function depends on three different hypergeometric functions of type 0F4. In Maple notation: a(n)=int( x^n*( 1/216*BesselK(1/3,2/3*sqrt(x))*(36*sqrt(3)*hypergeom([],[1/3, 4/3, 5/3, 2/3],1/243*x)*GAMMA(2/3)+8*3^(1/3)*x^(1/3)*Pi*hypergeom([],[2, 4/3, 5/3, 2/3],1/243*x)+3*3^(1/6)*GAMMA(2/3)^2*x^(2/3)*hypergeom([],[2, 7/3, 4/3, 5/3],1/243*x))/Pi/GAMMA(2/3)/exp(1)/x^(1/2) ), x=0..infinity), n=1,2....

a(2)=2!*LaguerreL(2,3,-1)=31, special value of associated Laguerre polynomial.

P. Blasiak, Karol A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Cf. A072020.

a[n_] := Sum[ 1/9*(3^n)^2 * Gamma[n + 1/3*k + 1/3] * Gamma[n + 1/3*k + 2/3] / Gamma[4/3 + 1/3*k ] / Gamma[5/3 + 1/3*k]/k!/Exp[1], {k, 0, Infinity}]
(* Second program: *)
a[n_] := Sum[Product[FactorialPower[k+3*(j-1), 2], {j, 1, n}]/k!, {k, 2, Infinity}]/E; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 16}] (* Jean-François Alcover, Sep 01 2016 *)

a(n) = Sum_{k=2..2*n} A091534(n, k) = (Sum_{k>=2} (1/k!)*Product_{j=1..n} fallfac(k+3*(j-1), 2))/exp(1), n>=1. From eq.(9) of the Blasiak et al. reference with r=5, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle). a(0) := 1 may be added.

Sum of an infinite series: a(n) = Sum_{k>=0} (1/9 * (3^n)^2 * Gamma(n+k/3+1/3) * Gamma(n+k/3+2/3) / (Gamma(4/3+k/3) * Gamma(5/3+k/3) * k! * exp(1))).

A091747 Generalized Stirling2 array (7,2).

Original entry on oeis.org

1, 42, 14, 1, 5544, 3192, 588, 42, 1, 1507968, 1165248, 321552, 41496, 2688, 84, 1, 696681216, 655966080, 232606080, 41497344, 4143552, 240240, 7980, 140, 1, 489070213632, 533531142144, 226306918656, 50249808000, 6575950080

Offset: 1

Wolfdieter Lang, Feb 27 2004

The sequence of row lengths 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.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

W. Lang, First 6 rows.

Cf. A078740 (3, 2)-Stirling2, A090438 (4, 2)-Stirling2, A091534 (5, 2)-Stirling2, A091746 (6, 2)-Stirling2.
Cf. A091545 (first column).
Cf. A091749 (row sums), A091751 (alternating row sums).

a(n, k)=(((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*product(fallfac(p+5*(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=7, s=2.

Recursion: a(n, k)=sum(binomial(2, p)*fallfac(5*(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=7, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle).

A092077 Generalized Stirling2 array (8,2).

Original entry on oeis.org

1, 56, 16, 1, 10192, 4928, 776, 48, 1, 3872960, 2477440, 575680, 63360, 3536, 96, 1, 2517424000, 1940556800, 572868800, 86163840, 7326880, 364800, 10480, 160, 1, 2497284608000, 2210343116800, 773352966400, 143430604800, 15836206400, 1099612800, 49056960, 1398400, 24520, 240, 1

Offset: 1

Wolfdieter Lang, Feb 27 2004

The sequence of row lengths 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, arXiv:quant-ph/0402027, 2004.
Wolfdieter Lang, First 6 rows.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

The generalized (k, 2)-Stirling2 arrays are, for k=2, ..., 7: A078739, A078740, A090438, A091534, A091746 and A091747.

Cf. A091546, A091552 (first, resp. second column). A091757 (row sums). A091758 (alternating row sums).

a[n_, k_] := ((-1)^k/k!) Sum[(-1)^p Binomial[k, p] Product[FactorialPower[ p + 6(j-1), 2], {j, 1, n}], {p, 2, k}];
Table[a[n, k], {n, 1, 6}, {k, 2, 2n}] // Flatten (* Jean-François Alcover, Feb 28 2020 *)

a(n, k) = (((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*product(fallfac(p+6*(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=8, s=2.

Recursion: a(n, k) = sum(binomial(2, p)*fallfac(6*(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=8, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle).

A091544 First column sequence of array A091746 ((6,2)-Stirling2).

Original entry on oeis.org

1, 30, 2700, 491400, 150368400, 69470200800, 45155630520000, 39285398552400000, 44078217175792800000, 61973973349164676800000, 106719182107261573449600000, 220908706962031457040672000000, 541226332056977069749646400000000, 1548989762347068373623487996800000000

Offset: 1

Wolfdieter Lang, Feb 13 2004

Also fifth column (m=4) sequence of triangle A091543.

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. A091535 (third column of A091543, first column of array A091534), A000407, A007696, A091746.

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

a(n) = 2^(n-1)*Product_{j=0..n-1}((2*j+1)*(4*j+1)), n>=1. From eq.12 of the Blasiak et al. reference with r=6, s=2, k=1.
a(n) = (2^(4*n-1))*risefac(1/4, n)*risefac(1/2, n), n>=1, with risefac(x, n) = Pochhammer(x, n).
a(n) = fac4(4*n-3)*fac4(4*n-2)/2, n>=1, with fac4(4*n-3) = A007696(n) and fac4(4*n-2)/2 = A000407(n+1) (quartic- or 4-factorials).
E.g.f.: (hypergeom([1/4, 1/2], [], 16*x)-1)/2.
a(n) = A091746(n, 2), n>=1.
a(n) ~ sqrt(Pi) * 2^(4*n) * n^(2*n-1/4) / (Gamma(1/4) * exp(2*n)). - Amiram Eldar, Aug 30 2025

cp's OEIS Frontend

A091535 First column (k=2) of array A091534 ((5,2)-Stirling2).

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

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

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

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

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A091746 Generalized Stirling2 array (6,2).

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A072019 Generalized Bell numbers B_{5,2}.

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A091747 Generalized Stirling2 array (7,2).

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A092077 Generalized Stirling2 array (8,2).

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