A101054 -id:A101054 - cp's OEIS frontend

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

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

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

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

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