A239301 -id:A239301 - cp's OEIS frontend

A121629 Finite sum involving signless Stirling numbers of the first kind and the Bell numbers. Appears in the process of normal ordering of n-th power of (a)^2(a+a), where a+ and a are boson creation and annihilation operators, respectively.

1, 3, 16, 121, 1179, 14026, 196783, 3177861, 58019356, 1181098459, 26515026561, 650572403218, 17316566815441, 496889918749251, 15288155067806104, 502024850361876481, 17522822345606176083, 647790109599863145106, 25283238154309049107231

Offset: 0

Karol A. Penson, Aug 12 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 0..350
K. A. Penson, P. Blasiak, A. Horzela, G. H. E. Duchamp and A. I. Solomon, Laguerre-type derivatives: Dobinski relations and combinatorial identities, J. Math. Phys. vol. 50, 083512 (2009)

Cf. A002720, A121630, A121631, A239301.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(((1-2*x)^(-1/2))-1)/(1-2*x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 17 2018

CoefficientList[Series[E^(((1-2*x)^(-1/2))-1)/(1-2*x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 29 2013 *)

x='x+O('x^30); Vec(serlaplace(exp(((1-2*x)^(-1/2))-1)/(1-2*x))) \\ G. C. Greubel, May 17 2018

a(n) = Sum_{p=1..n+1} abs(stirling1(n+1,p))*2^(n-p+1)*bell(p-1), n=0,1...
E.g.f.: exp(((1-2*x)^(-1/2))-1)/(1-2*x). - Vladeta Jovovic, Aug 13 2006
Recurrence: a(n) = (6*n-5)*a(n-1) - (2*n-3)*(6*n-7)*a(n-2) + 4*(2*n-3)*(n-2)^2*a(n-3). - Vaclav Kotesovec, Jun 29 2013
a(n) ~ 2^(n+5/6)*exp(3/2*(2*n)^(1/3)-1-n)*n^(n+1/3)/sqrt(3). - Vaclav Kotesovec, Jun 29 2013

Terms a(17) onward added by G. C. Greubel, May 17 2018

A121630 Finite sum involving signless Stirling numbers of the first kind and the Bell numbers. Appears in the process of normal ordering of n-th power of (a)^3(a+a), where a+ and a are boson creation and annihilation operators, respectively.

Original entry on oeis.org

1, 4, 29, 302, 4089, 68056, 1342949, 30635074, 792915057, 22952573484, 734630159341, 25757268041814, 981687991859689, 40407710444419072, 1786311057929722549, 84404172618241446506, 4244839086310722228449

Offset: 0

Karol A. Penson, Aug 12 2006

nonn

Cf. A002720, A121629, A121631, A239301.

CoefficientList[Series[E^(((1-3*x)^(-1/3))-1)/(1-3*x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Mar 14 2014 *)

a(n)=sum(abs(stirling1(n+1,p))*3^(n-p+1)*bell(p-1),p=1..n+1), n=0,1....
E.g.f.: exp(((1-3*x)^(-1/3))-1)/(1-3*x). - Vladeta Jovovic, Aug 13 2006
Recurrence: a(n) = 3*(4*n - 5)*a(n-1) - (54*n^2 - 189*n + 173)*a(n-2) + (108*n^3 - 729*n^2 + 1659*n - 1271)*a(n-3) - 9*(n-3)^2*(3*n - 8)*(3*n - 7)*a(n-4). - Vaclav Kotesovec, Mar 14 2014
a(n) ~ 1/2 * 3^(n+7/8) * exp(4*n^(1/4)/3^(3/4) - n - 1) * n^(n+3/8). - Vaclav Kotesovec, Mar 14 2014

A121631 Finite sum involving signless Stirling numbers of the first kind and the Bell numbers. Appears in the process of normal ordering of n-th power of (a)^4(a+a), where a+ and a are boson creation and annihilation operators, respectively.

Original entry on oeis.org

1, 5, 46, 613, 10679, 229576, 5868715, 173833661, 5853205468, 220767370219, 9219128625851, 422221005543250, 21041188313139901, 1133454896301865073, 65627299232007207934, 4064319309355535125201, 268077821490093243979235

Offset: 0

Karol A. Penson, Aug 12 2006

nonn

Cf. A002720, A121629, A121630, A239301.

CoefficientList[Series[E^(((1-4*x)^(-1/4))-1)/(1-4*x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Mar 14 2014 *)

a(n)=sum(abs(stirling1(n+1,p))*4^(n-p+1)*bell(p-1),p=1..n+1), n=0,1....
E.g.f.: exp(((1-4*x)^(-1/4))-1)/(1-4*x). - Vladeta Jovovic, Aug 13 2006
Recurrence: a(n) = 2*(10*n - 17)*a(n-1) - (160*n^2 - 704*n + 811)*a(n-2) + 2*(320*n^3 - 2592*n^2 + 7138*n - 6675)*a(n-3) - (1280*n^4 - 16384*n^3 + 79120*n^2 - 170816*n + 139079)*a(n-4) + 32*(n-4)^2*(2*n - 7)*(4*n - 15)*(4*n - 13)*a(n-5). - Vaclav Kotesovec, Mar 14 2014
a(n) ~ 1/sqrt(5) * 2^(2*n+9/5) * exp(5*n^(1/5)/2^(8/5)-n-1) * n^(n+2/5). - Vaclav Kotesovec, Mar 14 2014

cp's OEIS Frontend

A121629 Finite sum involving signless Stirling numbers of the first kind and the Bell numbers. Appears in the process of normal ordering of n-th power of (a)^2(a+a), where a+ and a are boson creation and annihilation operators, respectively.

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A121630 Finite sum involving signless Stirling numbers of the first kind and the Bell numbers. Appears in the process of normal ordering of n-th power of (a)^3(a+a), where a+ and a are boson creation and annihilation operators, respectively.

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A121631 Finite sum involving signless Stirling numbers of the first kind and the Bell numbers. Appears in the process of normal ordering of n-th power of (a)^4(a+a), where a+ and a are boson creation and annihilation operators, respectively.

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cp's OEIS Frontend

A121629 Finite sum involving signless Stirling numbers of the first kind and the Bell numbers. Appears in the process of normal ordering of n-th power of (a)^2*(a+*a), where a+ and a are boson creation and annihilation operators, respectively.

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Programs

Magma

Mathematica

PARI

Formula

Extensions

A121630 Finite sum involving signless Stirling numbers of the first kind and the Bell numbers. Appears in the process of normal ordering of n-th power of (a)^3*(a+*a), where a+ and a are boson creation and annihilation operators, respectively.

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Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A121631 Finite sum involving signless Stirling numbers of the first kind and the Bell numbers. Appears in the process of normal ordering of n-th power of (a)^4*(a+*a), where a+ and a are boson creation and annihilation operators, respectively.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A121629 Finite sum involving signless Stirling numbers of the first kind and the Bell numbers. Appears in the process of normal ordering of n-th power of (a)^2(a+a), where a+ and a are boson creation and annihilation operators, respectively.

A121630 Finite sum involving signless Stirling numbers of the first kind and the Bell numbers. Appears in the process of normal ordering of n-th power of (a)^3(a+a), where a+ and a are boson creation and annihilation operators, respectively.

A121631 Finite sum involving signless Stirling numbers of the first kind and the Bell numbers. Appears in the process of normal ordering of n-th power of (a)^4(a+a), where a+ and a are boson creation and annihilation operators, respectively.