A121630 -id:A121630 - 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

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).

A123333 a(n) = 3^n*(Gamma(n+1/3)/Gamma(1/3) + (n-1)!).

Original entry on oeis.org

4, 13, 82, 766, 9472, 145720, 2681200, 57411760, 1402226560, 38468725120, 1171102777600, 39174663404800, 1428249121868800, 56366281606835200, 2393966461645158400, 108871544042829568000

Offset: 1

Karol A. Penson, Sep 26 2006

nonn

The EXP-transform of a(n) is equal to A121630(n).

G. C. Greubel, Table of n, a(n) for n = 1..380

Cf. A121630.

With[{nmax = 50}, CoefficientList[Series[(1 - 3*x)^(-1/3) - 1 - Log[1 - 3*x], {x, 0, nmax}], x] Range[0, nmax]!] (* G. C. Greubel, Oct 12 2017 *)

x='x+O('x^30); Vec(serlaplace((1-3*x)^(-1/3) - 1 - log(1-3*x))) \\ G. C. Greubel, Oct 12 2017

E.g.f.: (1-3*x)^(-1/3) - 1 - log(1-3*x).

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

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A123333 a(n) = 3^n*(Gamma(n+1/3)/Gamma(1/3) + (n-1)!).

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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.

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).