A352270 -id:A352270 - cp's OEIS frontend

A101053 a(n) = n! * Sum_{k=0..n} Bell(k)/k! (cf. A000110).

1, 2, 6, 23, 107, 587, 3725, 26952, 219756, 1998951, 20105485, 221838905, 2666280457, 34689290378, 485840964614, 7288997427755, 116634438986227, 1982868327635663, 35692311974248093, 678159760252918824, 13563246929216611852, 284828660383365005643

Offset: 0

Karol A. Penson, Nov 29 2004

nonn

Sequence was originally defined as an infinite sum involving generalized Laguerre polynomials: a(n) = ((-1)^n*n!/exp(1))*Sum_{k>=0} LaguerreL(n,-n-1,k)/k!, n=0,1... . It appears in the problem of normal ordering of functions of boson operators.

a(n) is the number of ways to linearly order the elements in a (possibly empty) subset S of {1,2,...,n} and then partition the complement of S. - Geoffrey Critzer, Aug 07 2015

Robert Israel, Table of n, a(n) for n = 0..450

Cf. A000110, A186755, A278677, A352270.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(Exp(x)-1)/(1-x) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Mar 31 2019

with(combinat): a:=n->add(bell(j)*n!/j!,j=0..n): seq(a(n),n=0..20); # Zerinvary Lajos, Mar 19 2007

nn = 21; Range[0, nn]! CoefficientList[Series[Exp[(Exp[x]-1)]/(1-x), {x, 0, nn}], x] (* Geoffrey Critzer, Aug 07 2015 *)

egf(s)=my(v=Vec(s),i); while(polcoeff(s,i)==0,i++); i--; vector(i+#v,j,polcoeff(s,j+i)*(j+i)!)
egf(exp(exp(x)-1)/(1-x)) \\ Charles R Greathouse IV, Aug 07 2015

my(x='x+O('x^30)); Vec(serlaplace( exp(exp(x)-1)/(1-x) )) \\ G. C. Greubel, Mar 31 2019

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)*binomial(i-1, j-1)*v[i-j+1])); v; \\ Seiichi Manyama, Jul 14 2022

m = 30; T = taylor(exp(exp(x)-1)/(1-x), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Mar 31 2019

E.g.f: exp(exp(x)-1)/(1-x).
a(n) ~ exp(exp(1)-1) * n!. - Vaclav Kotesovec, Jun 26 2022
a(0) = 1; a(n) = Sum_{k=1..n} ((k-1)! + 1) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Jul 14 2022

New definition from Vladeta Jovovic, Dec 01 2004

A352271 Expansion of e.g.f. 1/(2 - exp(x) - log(1 + x)).

Original entry on oeis.org

1, 2, 8, 51, 427, 4485, 56461, 829619, 13929175, 263120293, 5522411441, 127497249825, 3211140897757, 87615489275587, 2574463431688695, 81050546853002151, 2721785052811891411, 97113737702073060713, 3668859532725782696709, 146306156466305491481253

Offset: 0

Seiichi Manyama, Mar 10 2022

nonn

Cf. A000670, A007840, A352147, A352270.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(2-exp(x)-log(1+x))))

a(n) = if(n==0, 1, sum(k=1, n, ((-1)^(k-1)*(k-1)!+1)*binomial(n, k)*a(n-k)));

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

A352292 Expansion of e.g.f. 1/(2 - exp(x) - x/(1 - x)).

Original entry on oeis.org

1, 2, 11, 91, 1007, 13941, 231645, 4490739, 99496787, 2480012329, 68684121713, 2092433179431, 69540117508119, 2503694594140845, 97076021030158565, 4032791843669289883, 178701570260701316219, 8413561430997560725713, 419425619946011214516345

Offset: 0

Seiichi Manyama, Mar 11 2022

nonn

Cf. A002866, A352270, A352293.

m = 18; Range[0, m]! * CoefficientList[Series[1/(2 - Exp[x] - x/(1 - x)), {x, 0, m}], x] (* Amiram Eldar, Mar 11 2022 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(2-exp(x)-x/(1-x))))

a(n) = if(n==0, 1, sum(k=1, n, (k!+1)*binomial(n, k)*a(n-k)));

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

cp's OEIS Frontend

A101053 a(n) = n! * Sum_{k=0..n} Bell(k)/k! (cf. A000110).

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A352271 Expansion of e.g.f. 1/(2 - exp(x) - log(1 + x)).

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PARI

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A352292 Expansion of e.g.f. 1/(2 - exp(x) - x/(1 - x)).

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Mathematica

PARI

PARI

Formula