A121629 -id:A121629 - cp's OEIS frontend

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.

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

A239301 E.g.f.: exp((1-5x)^(-1/5)-1)/(1-5x).

Original entry on oeis.org

1, 6, 67, 1090, 23265, 614302, 19323163, 705288522, 29296813825, 1364468928022, 70414831288275, 3987980655931570, 245910243177940897, 16399345182278307822, 1176033825828643912747, 90242683036826223141370, 7377887848681408224106497, 640225878087732419052020134

Offset: 0

Vaclav Kotesovec, Mar 14 2014

nonn

Generally, for e.g.f.: exp((1-p*x)^(-1/p)-1)/(1-p*x), and p>1, we have a(n) ~ 1/sqrt(p+1) * p^(n+(2*p+1)/(2*p+2)) * exp((p+1)*p^(-p/(p+1)) *n^(1/(p+1))-n-1) * n^(n+p/(2*p+2)).

Cf. A121629, A121630, A121631.

CoefficientList[Series[E^((1-5*x)^(-1/5)-1)/(1-5*x),{x,0,20}],x]*Range[0,20]!

a(n) = 5*(6*n - 13)*a(n-1) - 5*(75*n^2 - 400*n + 557)*a(n-2) + 50*(50*n^3 - 475*n^2 + 1539*n - 1698)*a(n-3) - (9375*n^4 - 137500*n^3 + 764625*n^2 - 1910000*n + 1807524)*a(n-4) + (18750*n^5 - 390625*n^4 + 3267500*n^3 - 13716875*n^2 + 28896490*n - 24436079)*a(n-5) - 25*(n-5)^2*(5*n - 24)*(5*n - 23)*(5*n - 22)*(5*n - 21)*a(n-6).

a(n) ~ 1/sqrt(6) * 5^(n+11/12) * exp(6*5^(-5/6)*n^(1/6)-n-1) * n^(n+5/12).

A123332 a(n) = 2^n*(Gamma(n+1/2)/Gamma(1/2) + (n-1)!).

Original entry on oeis.org

3, 7, 31, 201, 1713, 18075, 227295, 3317265, 55103265, 1026318195, 21181092975, 479733356025, 11829834687825, 315481555464075, 9046941599670975, 277598531343758625, 9075051786962786625, 314884420627497595875, 11557482238066613223375, 447385119579169194047625

Offset: 1

Karol A. Penson, Sep 26 2006

nonn

The EXP-transform of a(n) is equal to A121629(n).

G. C. Greubel, Table of n, a(n) for n = 1..400

Cf. A121629.

Table[2^n*(Gamma[n + 1/2]/Gamma[1/2] + (n - 1)!), {n, 0, 50}] (* G. C. Greubel, Oct 04 2017 *)

a(n) = round(2^n*(gamma(n+1/2)/gamma(1/2) + (n-1)!)); \\ Michel Marcus, Oct 05 2017

E.g.f.: (1-2*x)^(-1/2) - 1 - log(1-2*x).

cp's OEIS Frontend

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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Author

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Mathematica

Formula

A239301 E.g.f.: exp((1-5x)^(-1/5)-1)/(1-5x).

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Formula

A123332 a(n) = 2^n*(Gamma(n+1/2)/Gamma(1/2) + (n-1)!).

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cp's OEIS Frontend

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

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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

A239301 E.g.f.: exp((1-5*x)^(-1/5)-1)/(1-5*x).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A123332 a(n) = 2^n*(Gamma(n+1/2)/Gamma(1/2) + (n-1)!).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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.

A239301 E.g.f.: exp((1-5x)^(-1/5)-1)/(1-5x).