A101053 -id:A101053 - cp's OEIS frontend

A101054 E.g.f.: exp(exp(x)-1)/(1-x)^2.

1, 3, 12, 59, 343, 2302, 17537, 149711, 1417444, 14755947, 167664955, 2066153410, 27460121377, 391670868279, 5969233120520, 96827494235555, 1665874346755107, 30302732222472482, 581141491978752769, 11719848107849221435, 247960209086201040552, 5491993051193586857235

Offset: 0

Karol A. Penson, Nov 29 2004

nonn

Sequence appears in the problem of normal ordering of functions of boson operators.

G. C. Greubel, Table of n, a(n) for n = 0..447

Cf. A000110, A101053.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(Exp(x)-1)/(1-x)^2)); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 23 2018

egf := exp(exp(x)-1)/((1-x)^2);
seq(n!*coeff(series(egf,x,n+1),x,n),n=0..21); # Peter Luschny, Sep 17 2014

With[{nmax = 50}, CoefficientList[Series[Exp[Exp[x] - 1]/(1 - x)^2, {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, May 23 2018 *)

x='x+O('x^30); Vec(serlaplace(exp(exp(x)-1)/(1-x)^2 )) \\ G. C. Greubel, May 23 2018

a(n) = ((-1)^n*n!/e)*Sum_{k>=0} L(n,-n-2,k)/k!, where L is a generalized Laguerre polynomial.
a(n) = Sum_{k=0..n} binomial(n,k)*(k + 1)!*Bell(n-k), where Bell() = A000110. - Ilya Gutkovskiy, May 24 2018
a(n) ~ exp(exp(1)-1) * n * n!. - Vaclav Kotesovec, Jun 26 2022

A113059 a(n) = n! * Sum_{k=0..n} A000296(k)/k!.

Original entry on oeis.org

1, 1, 3, 10, 44, 231, 1427, 10151, 81923, 740732, 7425042, 81773715, 981864897, 12767876941, 178774288331, 2681781213130, 42909715480460, 729474427239587, 13130613291110603, 249482261007109579, 4989650444408388515, 104782705832468197252, 2305219956684224457858

Offset: 0

Karol A. Penson, Oct 12 2005

nonn

Number of set partitions of [n] where the k-th singletons are k-colored and all other blocks are unicolored. - Alois P. Heinz, Apr 29 2025

G. C. Greubel, Table of n, a(n) for n = 0..448

Cf. A000110, A000142, A000255, A000296, A101053, A101054.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(Exp(x)-1-x)/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 23 2018

b:= proc(n, k) option remember; `if`(n=0, 1, k*b(n-1, k+1)+
      add(b(n-j, k)*binomial(n-1, j-1), j=2..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 29 2025
# second Maple program:
b:= proc(n, k, m) option remember; `if`(n=0, k!, `if`(k>0,
      b(n-1, k-1, m+1)*k, 0)+m*b(n-1, k, m)+b(n-1, k+1, m))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 29 2025

With[{nmax = 50}, CoefficientList[Series[Exp[Exp[x] - 1 - x]/(1 - x), {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, May 23 2018 *)

x='x+O('x^30); Vec(serlaplace( exp(exp(x)-1-x)/(1-x))) \\ G. C. Greubel, May 23 2018

a(n) = (-1)^n*n!*Sum_{k >=0} LaguerreL(n, -n-1, k-1)/k!/exp(1), n>=0.
E.g.f.: exp(exp(x)-1-x)/(1-x).
a(n) ~ exp(exp(1)-2) * n!. - Vaclav Kotesovec, Jun 26 2022

A101055 E.g.f.: exp(exp(x)-1)/(1-x)^3.

Original entry on oeis.org

1, 4, 20, 119, 819, 6397, 55919, 541144, 5746596, 66475311, 832418065, 11222752125, 162133146877, 2499401777680, 40960858008040, 711240364356155, 13045720176453587, 252079975222183461, 5118581045978055067, 108972887981432267708, 2427417968714846394712, 56467770394205361146187

Offset: 0

Karol A. Penson, Nov 29 2004

nonn

Sequence appears in the problem of normal ordering of functions of boson operators.

Harvey P. Dale, Table of n, a(n) for n = 0..447
Index entries for sequences related to Laguerre polynomials

Cf. A000110, A101053, A101054.

With[{nn=30},CoefficientList[Series[Exp[Exp[x]-1]/(1-x)^3,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 11 2021 *)

a(n) = ((-1)^n*n!/e)*Sum_{k>=0} L(n,-n-3,k)/k!, where L is a generalized Laguerre polynomial.
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*(k + 2)!*Bell(n-k), where Bell() = A000110. - Ilya Gutkovskiy, May 24 2018
a(n) ~ exp(exp(1)-1) * n^2 * n! / 2. - Vaclav Kotesovec, Jun 26 2022

Terms after a(15) from Ilya Gutkovskiy, May 24 2018

A367972 Expansion of e.g.f. exp(exp(2*x) - 1)/(1 - x).

Original entry on oeis.org

1, 3, 14, 82, 568, 4504, 40016, 392368, 4198784, 48616320, 604921600, 8043848960, 113785080832, 1705669278720, 27007064393728, 450422751508480, 7893590619881472, 145052304752934912, 2789743827826573312, 56063169473909817344

Offset: 0

Seiichi Manyama, Dec 06 2023

nonn

Cf. A101053, A367971.

Cf. A000110, A367973.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, ((j-1)!+2^j)*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} ((k-1)! + 2^k) * binomial(n-1,k-1) * a(n-k).

a(n) = n! * Sum_{k=0..n} 2^k * Bell(k)/k!, where Bell() is A000110.

A305051 a(n) = n! * [x^n] exp(exp(x) - 1)/(1 - x)^n.

Original entry on oeis.org

1, 2, 12, 119, 1655, 29647, 649925, 16852656, 504519916, 17124927207, 649856846635, 27262957861405, 1252893494644357, 62593349657218070, 3377648236341185084, 195782612085816693995, 12131925601060324633027, 800321307922970722566527, 55998398887720317868148977

Offset: 0

Ilya Gutkovskiy, May 24 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..360
Eric Weisstein's World of Mathematics, Bell Number
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Cf. A000110, A101053, A101054, A101055.

Table[n! SeriesCoefficient[Exp[Exp[x] - 1]/(1 - x)^n, {x, 0, n}], {n, 0, 18}]
Table[(-1)^n n!/Exp[1] Sum[LaguerreL[n,-2 n, k]/k!, {k, 0, Infinity}], {n, 0, 18}]
Join[{1}, Table[1/(n - 1)! Sum[Binomial[n, k] (n + k - 1)! BellB[n - k], {k, 0, n}], {n, 18}]]

a(n) = ((-1)^n*n!/exp(1))*Sum_{k>=0} Laguerre(n,-2*n,k)/k!.
a(0) = 1; a(n) = (1/(n - 1)!)*Sum_{k=0..n} binomial(n,k)*(n + k - 1)!*Bell(n-k), where Bell() = A000110.
a(n) ~ c * n^n * 4^n / exp(n), where c = exp(exp(1/2) - 1)/sqrt(2) = 1.3527609882698012767793757868699146219161180684881726130481416807461987206887... - Vaclav Kotesovec, May 11 2021, updated Mar 18 2024

A317366 Expansion of e.g.f. exp(exp(x/(1 - x)) - 1)/(1 - x).

Original entry on oeis.org

1, 2, 8, 47, 359, 3347, 36665, 460098, 6494444, 101708007, 1748263435, 32697711895, 660642793717, 14332871438810, 332186039584768, 8188070581358795, 213821204277955267, 5895325327054011087, 171095582314380667621, 5212792218964517899506, 166321395872186089502972, 5545223090189205308551443

Offset: 0

Ilya Gutkovskiy, Jul 26 2018

nonn

Cf. A000110, A002720, A084357, A101053.

a:= proc(n) option remember; add(binomial(n, k)^2
      *k!*combinat[bell](n-k), k=0..n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 26 2018

nmax = 21; CoefficientList[Series[Exp[Exp[x/(1 - x)] - 1]/(1 - x) , {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k]^2 k! BellB[n - k], {k, 0, n}], {n, 0, 21}]

a(n) = Sum_{k=0..n} binomial(n,k)^2*k!*Bell(n-k), where Bell() = A000110.

A367971 Expansion of e.g.f. exp(exp(-x) - 1)/(1 - x).

Original entry on oeis.org

1, 0, 2, 1, 19, 43, 461, 2350, 22940, 185313, 1969105, 20981585, 255992617, 3300259584, 46394533498, 694535043925, 11123040844947, 189008829494295, 3402841007703469, 64648146404160854, 1293014652241452452, 27152832827254344741, 597366828915334031625

Offset: 0

Seiichi Manyama, Dec 06 2023

nonn

Cf. A101053, A367972.

Cf. A000110, A087650, A186755.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, ((j-1)!+(-1)^j)*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} ((k-1)! + (-1)^k) * binomial(n-1,k-1) * a(n-k).

a(n) = n! * Sum_{k=0..n} (-1)^k * Bell(k)/k!, where Bell() is A000110.

A337000 E.g.f.: 1 / ((1 - x)*(2 - exp(x))).

Original entry on oeis.org

1, 2, 7, 34, 211, 1596, 14259, 147106, 1722683, 22591408, 328161643, 5232410646, 90880495347, 1708304787892, 34557610000931, 748647340991818, 17294012137850443, 424368973372593432, 11024176183963527099, 302260934814635426014, 8722906492537092723395

Offset: 0

Ilya Gutkovskiy, Aug 10 2020

nonn

Exponential convolution of Fubini numbers (A000670) and factorial numbers (A000142).

Cf. A000142, A000670, A101053.

nmax = 20; CoefficientList[Series[1/((1 - x) (2 - Exp[x])), {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[HurwitzLerchPhi[1/2, -k, 0]/(2 k!), {k, 0, n}], {n, 0, 20}]
a[0] = 1; a[n_] := a[n] = 2 n a[n - 1] - Sum[Binomial[n, k]  (k - 1) a[n - k], {k, 2, n}]; Table[a[n], {n, 0, 20}]

a(n) = n! * Sum_{k=0..n} A000670(k) / k!.
a(0) = 1; a(n) = 2 * n * a(n-1) - Sum_{k=2..n} binomial(n,k) * (k-1) * a(n-k).
a(n) ~ n! / (2 * (1 - log(2)) * log(2)^(n+1)). - Vaclav Kotesovec, Aug 09 2021

A356559 a(n) = exp(-1) * n! * Sum_{k>=0} Laguerre(n,k) / k!.

Original entry on oeis.org

1, 0, 0, 1, 7, 43, 281, 2056, 17004, 157809, 1622515, 18245335, 222004597, 2898508416, 40343356184, 595578837205, 9287308741827, 152459628788599, 2627373030049669, 47425289731038656, 895098852673047772, 17644305594671247141, 363065584549610882703, 7799894520723959486795

Offset: 0

Ilya Gutkovskiy, Aug 12 2022

nonn

Eric Weisstein's World of Mathematics, Laguerre Polynomial

Cf. A000110, A009940, A101053, A317362, A317366.

Table[Exp[-1] n! Sum[LaguerreL[n, k]/k!, {k, 0, Infinity}], {n, 0, 23}]
nmax = 23; CoefficientList[Series[Exp[Exp[-x/(1 - x)] - 1]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) Binomial[n, k]^2 k! BellB[n - k], {k, 0, n}], {n, 0, 23}]

my(x='x+O('x^30)); Vec(serlaplace(exp(exp(-x/(1 - x)) - 1) / (1 - x))) \\ Michel Marcus, Aug 12 2022

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

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k)^2 * k! * Bell(n-k).

cp's OEIS Frontend

A101054 E.g.f.: exp(exp(x)-1)/(1-x)^2.

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A113059 a(n) = n! * Sum_{k=0..n} A000296(k)/k!.

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A101055 E.g.f.: exp(exp(x)-1)/(1-x)^3.

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A367972 Expansion of e.g.f. exp(exp(2*x) - 1)/(1 - x).

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A305051 a(n) = n! * [x^n] exp(exp(x) - 1)/(1 - x)^n.

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A317366 Expansion of e.g.f. exp(exp(x/(1 - x)) - 1)/(1 - x).

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A367971 Expansion of e.g.f. exp(exp(-x) - 1)/(1 - x).

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A337000 E.g.f.: 1 / ((1 - x)*(2 - exp(x))).

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