A123511 -id:A123511 - cp's OEIS frontend

A123510 Arises in the normal ordering of functions of a(a+)a, where a and a+ are the boson annihilation and creation operators, respectively.

1, 6, 42, 340, 3135, 32466, 373156, 4713192, 64877805, 966466270, 15487707246, 265617899196, 4853435351947, 94114052406570, 1930026941433480, 41728495237790416, 948549349736725401, 22613209058160908982, 564104540143144909810, 14694713818659640322340

Offset: 0

Karol A. Penson, Oct 02 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 0..440

Cf. A002720, A052852, A123511, A123512.

I:=[6,42]; [1] cat [n le 2 select I[n] else 2*(n+2)*Self(n-1) - (n^2 -1)*((n+2)/n)*Self(n-2): n in [1..30]]; // G. C. Greubel, May 16 2018

max = 16; s = (1/(1-x)^3)*Exp[x/(1-x)]*LaguerreL[2, -x/(1-x)] + O[x]^(max+1); CoefficientList[s, x]*Range[0, max]! (* Jean-François Alcover, May 23 2016 *)

m=30; v=concat([6,42], vector(m-2)); for(n=3, m, v[n]=2*(n+2)*v[n-1]-(n^2 - 1)*((n+2)/n)*v[n-2]); concat([1], v) \\ G. C. Greubel, May 16 2018

E.g.f.: (1/(1-x)^3)*exp(x/(1-x))*LaguerreL(2,-x/(1-x)), where LaguerreL(p,y) are the Laguerre polynomials.
From Vaclav Kotesovec, Nov 13 2017: (Start)
Recurrence: n*a(n) = 2*n*(n+2)*a(n-1) - (n-1)*(n+1)*(n+2)*a(n-2).
a(n) ~ exp(2*sqrt(n) - n - 1/2) * n^(n + 9/4) / 2^(3/2) * (1 + 31/(48*sqrt(n))).
(End)

a(0)=1 prepended by G. C. Greubel, Oct 31 2017

More terms from G. C. Greubel, May 16 2018

A123512 Arises in the normal ordering of functions of a(a+)a, where a and a+ are the boson annihilation and creation operators, respectively.

Original entry on oeis.org

1, 10, 105, 1190, 14630, 194796, 2798670, 43204260, 713655855, 12564061510, 234896893231, 4648313235930, 97068707038940, 2133251854548920, 49215687006553740, 1189262114277026856, 30037396074996304365

Offset: 0

Karol A. Penson, Oct 02 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 0..439

Cf.: A002720, A052852, A123510, A123511.

CoefficientList[ Series[(1/(1 - x)^5)*Exp[x/(1 - x)]LaguerreL[4, -x/(1 - x)], {x,0,16}], x]*Range[0, 16]! (* Robert G. Wilson v, Oct 03 2006 *)

LaguerreL(n,v='x) = {
  my(x='x+O('x^(n+1)), t='t);
  subst(polcoeff(exp(-x*t/(1-x))/(1-x), n), 't, v);
};
N=17;x='x+O('x^N); Vec(serlaplace((1/(1-x)^5)*exp(x/(1-x))*LaguerreL(4,-x/(1-x)))) \\ Gheorghe Coserea, Oct 26 2017

E.g.f.: (1/(1-x)^5)*exp(x/(1-x))*LaguerreL(4,-x/(1-x)).
From Vaclav Kotesovec, Nov 13 2017: (Start)
Recurrence: n*a(n) = 2*n*(n+4)*a(n-1) - (n-1)*(n+3)*(n+4)*a(n-2).
a(n) ~ exp(2*sqrt(n)-n-1/2) * n^(n + 17/4) / (3*2^(7/2)) * (1 + 31/(48*sqrt(n))).
(End)

a(0)=1 prepended by Gheorghe Coserea, Oct 26 2017

A123525 Arises in the normal ordering of functions of a(a+)a, where a and a+ are the boson annihilation and creation operators, respectively.

Original entry on oeis.org

2, 14, 102, 836, 7730, 79962, 916454, 11533832, 158149026, 2346622310, 37458934502, 640013453004, 11652216012242, 225169809833906, 4602407562991590, 99194703240441872

Offset: 1

Karol A. Penson, Oct 02 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 1..440

Cf.: A002720, A052852, A123510, A123511, A123512, A123526, A123527.

Rest[With[{nmax = 50}, CoefficientList[Series[(1/(1 - x)^2)*Exp[x/(1 - x)]*LaguerreL[1, 1/(x - 1)]*x, {x, 0, nmax}], x]*Range[0, nmax]!]] (* G. C. Greubel, Oct 14 2017 *)

E.g.f.: (1/(1-x)^2)*exp(x/(1-x))*LaguerreL(1,1/(x-1))*x.
From Vaclav Kotesovec, Nov 13 2017: (Start)
Recurrence: (n-2)*(n-1)*a(n) = 2*(n-2)*n^2*a(n-1) - (n-1)^3*n*a(n-2).
a(n) ~ exp(2*sqrt(n) - n - 1/2) * n^(n + 5/4) / sqrt(2) * (1 + 31/(48*sqrt(n))).
(End)

A123686 E.g.f.: (1-x^4)^(-1/2)exp(x^2/(1-x^2))BesselI(0,x^2/(x^2-1)) (since this is an even function, we do not give the intercalating 0's).

Original entry on oeis.org

1, 2, 54, 2460, 239190, 33124140, 6896500380, 1879519201560, 674900483206950, 300426422192196300, 164868151446145847700, 108046627817926248851400, 83890281074290204071858300, 75722368306901033144261835000

Offset: 0

Karol A. Penson, Oct 06 2006

nonn

Arises in the normal ordering of functions of a*(a+)*a, where a and a+ are the boson annihilation and creation operators, respectively.

G. C. Greubel, Table of n, a(n) for n = 0..220

Cf. A123510, A123511, A123512, A123525.

G:=(1-x^4)^(-1/2)*exp(x^2/(1-x^2))*BesselI(0,x^2/(x^2-1)): Gser:=series(G,x=0,40): seq((2*n)!*coeff(Gser,x,2*n),n=0..15); # Emeric Deutsch, Oct 31 2006

With[{nmax = 50}, CoefficientList[Series[(1 - x^4)^(-1/2)*Exp[x^2/(1 - x^2)]*BesselI[0, x^2/(x^2 - 1)], {x, 0, nmax}], x]*Range[0, nmax]!][[;; ;; 2 ]] (* G. C. Greubel, Oct 18 2017 *)

From Vaclav Kotesovec, Nov 13 2017: (Start)
Recurrence: n*a(n) = 2*(2*n - 1)*(3*n - 2)*a(n-1) + 4*(n-1)^2*(2*n - 3)*(2*n - 1)*(2*n + 1)*a(n-2) + 8*(n-2)^2*(n-1)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-3) - 16*(n-3)^2*(n-2)^2*(n-1)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-4).
a(n) ~ 2^(2*n - 3/4) * exp(2*sqrt(2*n) - 2*n -1) * n^(2*n - 1/4) / sqrt(Pi) * (1 + 91/(48*sqrt(2*n))). (End)

More terms from Emeric Deutsch, Oct 31 2006

A123687 E.g.f.: (1-x^2)^(-1/2)exp(x^2/(1-x^2))BesselI(0,x^2/(x^2-1)) (since this is an even function, we do not give the intercalating 0's).

Original entry on oeis.org

1, 3, 63, 3225, 297675, 42805665, 8790957945, 2433297161295, 870928551367875, 390718610250593625, 214426984078881899325, 141173178618822867992475, 109729771971447612972712725, 99352716603692210781106359375

Offset: 0

Karol A. Penson, Oct 06 2006

nonn

Robert Israel, Table of n, a(n) for n = 0..221

Cf. A123510, A123511, A123512, A123525, A123686.

G:=(1-x^2)^(-1/2)*exp(x^2/(1-x^2))*BesselI(0,x^2/(x^2-1)): Gser:=series(G,x=0,40): seq((2*n)!*coeff(Gser,x,2*n),n=0..15); # Emeric Deutsch, Oct 31 2006

DeleteCases[Flatten@ MapIndexed[#1 (#2 - 1)! &, CoefficientList[Series[(1 - x^2)^(-1/2) Exp[x^2/(1 - x^2)] BesselI[0, x^2/(x^2 - 1)], {x, 0, 26}], x]], 0] (* Michael De Vlieger, Oct 10 2016 *)
With[{nmax = 50}, CoefficientList[Series[(1 - x^2)^(-1/2)*Exp[x^2/(1 - x^2)]*BesselI[0, x^2/(x^2 - 1)], {x, 0, nmax}], x]*Range[0, nmax]!][[;; ;; 2 ]] (* G. C. Greubel, Oct 18 2017 *)

(n+1)*(2*n+3)^2*(2*n+1)^2*a(n) - (2*n+5)*(2*n+3)^2*a(n+1) + (n+2)*a(n+2) = 0. - Robert Israel, Oct 10 2016

a(n) ~ 2^(2*n - 1/4) * exp(2*sqrt(2*n) - 2*n - 1) * n^(2*n - 1/4) / sqrt(Pi) * (1 + 67/(48*sqrt(2*n))). - Vaclav Kotesovec, Nov 13 2017

More terms from Emeric Deutsch, Oct 31 2006

cp's OEIS Frontend

A123510 Arises in the normal ordering of functions of a*(a+)*a, where a and a+ are the boson annihilation and creation operators, respectively.

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A123512 Arises in the normal ordering of functions of a*(a+)*a, where a and a+ are the boson annihilation and creation operators, respectively.

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A123525 Arises in the normal ordering of functions of a*(a+)*a, where a and a+ are the boson annihilation and creation operators, respectively.

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A123686 E.g.f.: (1-x^4)^(-1/2)*exp(x^2/(1-x^2))*BesselI(0,x^2/(x^2-1)) (since this is an even function, we do not give the intercalating 0's).

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A123687 E.g.f.: (1-x^2)^(-1/2)*exp(x^2/(1-x^2))*BesselI(0,x^2/(x^2-1)) (since this is an even function, we do not give the intercalating 0's).

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Maple

Mathematica

Formula

Extensions

A123510 Arises in the normal ordering of functions of a(a+)a, where a and a+ are the boson annihilation and creation operators, respectively.

A123512 Arises in the normal ordering of functions of a(a+)a, where a and a+ are the boson annihilation and creation operators, respectively.

A123525 Arises in the normal ordering of functions of a(a+)a, where a and a+ are the boson annihilation and creation operators, respectively.

A123686 E.g.f.: (1-x^4)^(-1/2)exp(x^2/(1-x^2))BesselI(0,x^2/(x^2-1)) (since this is an even function, we do not give the intercalating 0's).

A123687 E.g.f.: (1-x^2)^(-1/2)exp(x^2/(1-x^2))BesselI(0,x^2/(x^2-1)) (since this is an even function, we do not give the intercalating 0's).