A201354 -id:A201354 - cp's OEIS frontend

A201365 Expansion of e.g.f. exp(x) / (5 - 4*exp(x)).

1, 5, 45, 605, 10845, 243005, 6534045, 204972605, 7348546845, 296387331005, 13282361478045, 654762261324605, 35211177242722845, 2051349014835939005, 128701394409842982045, 8651475271312083756605, 620334325261670875138845, 47259638324026516284867005

Offset: 0

Paul D. Hanna, Nov 30 2011

			E.g.f.: E(x) = 1 + 5*x + 45*x^2/2! + 605*x^3/3! + 10845*x^4/4! + 243005*x^5/5! + ...
O.g.f.: A(x) = 1 + 5*x + 45*x^2 + 605*x^3 + 10845*x^4 + 243005*x^5 + ...
where A(x) = 1 + 5*x/(1+x) + 2!*5^2*x^2/((1+x)*(1+2*x)) + 3!*5^3*x^3/((1+x)*(1+2*x)*(1+3*x)) + 4!*5^4*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) + ...

G. C. Greubel, Table of n, a(n) for n = 0..358

Cf. A027882, A032183, A050353.
Cf. A201366, A201367, A201368.
Cf. A000629, A201339, A201354.

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!(Laplace( 1/(5*Exp(-x) -4) ))); // G. C. Greubel, Jun 08 2020

seq(coeff(series(1/(5*exp(-x) - 4), x, n+1)*n!, x, n), n = 0..20); # G. C. Greubel, Jun 08 2020

Table[Sum[(-1)^(n-k)*5^k*StirlingS2[n,k]*k!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 13 2013 *)
With[{nn=20},CoefficientList[Series[Exp[x]/(5-4Exp[x]),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jul 09 2015 *)
a[n_]:= If[n<0, 0, PolyLog[ -n, 4/5]/4]; (* Michael Somos, Apr 27 2019 *)

{a(n)=n!*polcoeff(exp(x+x*O(x^n))/(5 - 4*exp(x+x*O(x^n))), n)}

{a(n)=polcoeff(sum(m=0, n, 5^m*m!*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)}

{a(n)=sum(k=0, n, (-1)^(n-k)*5^k*stirling(n, k, 2)*k!)}

[sum( (-1)^(n-j)*5^j*factorial(j)*stirling_number2(n,j) for j in (0..n)) for n in (0..20)] # G. C. Greubel, Jun 08 2020

O.g.f.: A(x) = Sum_{n>=0} n! * 5^n*x^n / Product_{k=0..n} (1+k*x).
O.g.f.: A(x) = 1/(1 - 5*x/(1-4*x/(1 - 10*x/(1-8*x/(1 - 15*x/(1-12*x/(1 - 20*x/(1-16*x/(1 - 25*x/(1-20*x/(1 - ...))))))))))), a continued fraction.
a(n) = Sum_{k=0..n} (-1)^(n-k) * 5^k * Stirling2(n,k) * k!.
a(n) = Sum_{k=0..n} A123125(n,k)*5^k*4^(n-k). - Philippe Deléham, Nov 30 2011
a(n) ~ n! / (4*(log(5/4))^(n+1)) . - Vaclav Kotesovec, Jun 13 2013
a(n) = log(5/4) * Integral_{x = 0..oo} (ceiling(x))^n * (5/4)^(-x) dx. - Peter Bala, Feb 14 2015
a(n) = (1/4) Sum_{k>=1} (4/5)^k * n^k. - Michael Somos, Apr 27 2019
a(n) = 1 + 4 * Sum_{k=0..n-1} binomial(n,k) * a(k). - Ilya Gutkovskiy, Jun 08 2020
From Seiichi Manyama, Nov 15 2023: (Start)
a(0) = 1; a(n) = -5*Sum_{k=1..n} (-1)^k * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 5*a(n-1) + 4*Sum_{k=1..n-1} binomial(n-1,k) * a(n-k). (End)
a(n) = (5/4)*A094417(n) - (1/4)*0^n. - Seiichi Manyama, Dec 21 2023

A278075 Coefficients of the signed Fubini polynomials in ascending order, F_n(x) = Sum_{k=0..n} (-1)^nStirling2(n,k)k!*(-x)^k.

Original entry on oeis.org

1, 0, 1, 0, -1, 2, 0, 1, -6, 6, 0, -1, 14, -36, 24, 0, 1, -30, 150, -240, 120, 0, -1, 62, -540, 1560, -1800, 720, 0, 1, -126, 1806, -8400, 16800, -15120, 5040, 0, -1, 254, -5796, 40824, -126000, 191520, -141120, 40320, 0, 1, -510, 18150, -186480, 834120, -1905120, 2328480, -1451520, 362880

Offset: 0

Peter Luschny, Jan 09 2017

sign

Signed version of A131689.

Integral_{x=0..1} F_n(x) = B_n(1) where B_n(x) are the Bernoulli polynomials.

			Triangle of coefficients starts:
[1]
[0,  1]
[0, -1,    2]
[0,  1,   -6,    6]
[0, -1,   14,  -36,    24]
[0,  1,  -30,  150,  -240,   120]
[0, -1,   62, -540,  1560, -1800,    720]
[0,  1, -126, 1806, -8400, 16800, -15120, 5040]

Peter Luschny, Illustration of the polynomials.
Peter Luschny, The Bernoulli Manifesto.
Grzegorz Rządkowski, Bernoulli numbers and solitons - revisited, Journal of Nonlinear Mathematical Physics, (2010) 17:1, 121-126.
J. Worpitzky, Studien über die Bernoullischen und Eulerschen Zahlen, Journal für die reine und angewandte Mathematik, 94 (1883), 203-232.

Row sums are A000012, diagonal is A000142.
Cf. A131689 (unsigned), A019538 (n>0, k>0), A090582.
Let F(n, x) = Sum_{k=0..n} T(n,k)*x^k then, apart from possible differences in the sign or the offset, we have: F(n, -5) = A094418(n), F(n, -4) = A094417(n), F(n, -3) = A032033(n), F(n, -2) = A004123(n), F(n, -1) = A000670(n), F(n, 0) = A000007(n), F(n, 1) = A000012(n), F(n, 2) = A000629(n), F(n, 3) = A201339(n), F(n, 4) = A201354(n), F(n, 5) = A201365(n).
Cf. A163626, A173018.

function T(n, k)
    if k < 0 || k > n return 0 end
    if n == 0 && k == 0 return 1 end
    k*(T(n-1, k-1) - T(n-1, k))
end
for n in 0:7
    println([T(n,k) for k in 0:n])
end
# Peter Luschny, Mar 26 2020

F := (n,x) -> add((-1)^n*Stirling2(n,k)*k!*(-x)^k, k=0..n):
for n from 0 to 10 do PolynomialTools:-CoefficientList(F(n,x), x) od;

T[ n_, k_] := If[ n < 0 || k < 0, 0, (-1)^(n - k) k! StirlingS2[n, k]]; (* Michael Somos, Jul 08 2018 *)

{T(n, k) = if( n<0, 0, sum(i=0, k, (-1)^(n + i) * binomial(k, i) * i^n))};
/* Michael Somos, Jul 08 2018 */

T(n, k) = (-1)^(n-k) * Stirling2(n, k) * k!.
E.g.f.: 1/(1-x*(1-exp(-t))) = Sum_{n>=0} F_n(x) t^n/n!.
T(n, k) = k*(T(n-1, k-1) - T(n-1, k)) for 0 <= k <= n, T(0, 0) = 1, otherwise 0.
Bernoulli numbers are given by B(n) = Sum_{k = 0..n} T(n, k) / (k+1) with B(1) = 1/2. - Michael Somos, Jul 08 2018
Let F_n(x) be the row polynomials of this sequence and W_n(x) the row polynomials of A163626. Then F_n(1 - x) = W_n(x) and Integral_{x=0..1} U(n, x) = Bernoulli(n, 1) for U in {W, F}. - Peter Luschny, Aug 10 2021
T(n, k) = [z^k] Sum_{k=0..n} Eulerian(n, k)*z^(k+1)*(z-1)^(n-k-1) for n >= 1, where Eulerian(n, k) = A173018(n, k). - Peter Luschny, Aug 15 2022

A090355 G.f. satisfies A^4 = BINOMIAL(A)^3.

Original entry on oeis.org

1, 3, 15, 109, 1086, 14178, 232906, 4647006, 109376595, 2967406345, 91130074437, 3123199831983, 118106517900868, 4883161763750820, 219076867059030300, 10597531747143624820, 549768536732090716371, 30443800514118532762329

Offset: 0

Paul D. Hanna, Nov 26 2003

See comments in A090353.

Cf. A090353, A090354; A019538, A032033, A050352, A201354.

Cf. A090352, A090357, A090362.

nmax = 17; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x]^4 - A[x/(1 - x)]^3/(1 - x)^3 + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 02 2019 *)

{a(n)=local(A); if(n<1,0,A=1+x+x*O(x^n); for(k=1,n,B=subst(A,x,x/(1-x))/(1-x)+x*O(x^n); A=A-A^4+B^3);polcoeff(A,n,x))}

G.f.: A(x)^4 = A(x/(1-x))^3/(1-x)^3.
From Peter Bala, May 26 2015: (Start)
O.g.f.: A(x) = exp( Sum_{n >= 1} b(n)*x^n/n ), where b(n) = Sum_{k = 1..n} k!*Stirling2(n,k)*3^k = A032033(n) = 3*A050352(n).
BINOMIAL(A(x)) = exp( Sum_{n >= 1} c(n)*x^n/n ) where c(n) = (-1)^n*Sum_{k = 1..n} k!*Stirling2(n,k)*4^k = A201354(n) = 4*A050352(n) for n >= 1. A(x) = B(x)^3 and BINOMIAL(A(x)) = B(x)^4 where B(x) = 1 + x + 4*x^2 + 28*x^3 + 286*x^4 + ... is the o.g.f. for A090353. See also A019538. (End)
G.f.: Product_{k>=1} 1/(1 - k*x)^((1/4) * (3/4)^k). - Seiichi Manyama, May 26 2025
a(n) ~ (n-1)! / (4 * log(4/3)^(n+1)). - Vaclav Kotesovec, May 28 2025

A136728 E.g.f.: A(x) = (exp(x)/(4 - 3*exp(x)))^(1/4).

Original entry on oeis.org

1, 1, 4, 31, 349, 5146, 93799, 2036161, 51283894, 1470035101, 47250248569, 1683031711516, 65800765032589, 2801364476781781, 129003301751229364, 6389120632590635971, 338644807090096148809, 19126604338708282552186

Offset: 0

Paul D. Hanna, Jan 24 2008

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..373

Cf. A201354, variants: A014307, A136727, A136729.

CoefficientList[Series[(E^x/(4-3*E^x))^(1/4), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 15 2013 *)

a(n)=n!*polcoeff((exp(x +x*O(x^n))/(4-3*exp(x +x*O(x^n))))^(1/4),n)

/* As solution to integral equation: */ a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+intformal(A^4*exp(-x+x*O(x^n))));n!*polcoeff(A,n)

E.g.f. A(x) satisfies: A(x) = 1 + integral( A(x)^5 * exp(-x) ).
O.g.f.: 1/(1 - x/(1-3*x/(1 - 5*x/(1-6*x/(1 - 9*x/(1-9*x/(1 - 13*x/(1-12*x/(1 - 17*x/(1-15*x/(1 - ...))))))))))), a continued fraction.
G.f.: 1/G(0) where G(k) = 1 - x*(4*k+1)/( 1 - 3*x*(k+1)/G(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 23 2013
a(n) ~ n! * Gamma(3/4)/(sqrt(2)*3^(1/4)*n^(3/4)*Pi*log(4/3)^(n+1/4)). - Vaclav Kotesovec, Jun 15 2013
a(n) = 1 + 3 * Sum_{k=1..n-1} (binomial(n,k) - 1) * a(k). - Ilya Gutkovskiy, Jul 09 2020
From Seiichi Manyama, Nov 15 2023: (Start)
a(n) = Sum_{k=0..n} (-1)^(n-k) * (Product_{j=0..k-1} (4*j+1)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^k * (3*k/n - 4) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = a(n-1) + 3*Sum_{k=1..n-1} binomial(n-1,k) * a(n-k). (End)

A343709 a(n) = 1 + 3 * Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).

Original entry on oeis.org

1, 4, 28, 295, 4159, 73348, 1552468, 38336569, 1081926157, 34350646636, 1211796777748, 47023762576987, 1990643657768683, 91291802205304972, 4508735102829489580, 238583762726054522989, 13466532093135977880025, 807606110028529741369396, 51282242176105846536128236

Offset: 0

Ilya Gutkovskiy, Apr 26 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..371

Cf. A201354, A291979, A343707, A343710.

a[n_] := a[n] = 1 + 3 Sum[Binomial[n, k] (n - k - 1)! a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[x]/(1 + 3 Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!

N=20; x='x+O('x^N); Vec(serlaplace(exp(x)/(1+3*log(1-x)))) \\ Seiichi Manyama, Oct 20 2021

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

A201355 Expansion of e.g.f.: 3exp(3x) / (4 - exp(3*x)).

Original entry on oeis.org

1, 4, 20, 132, 1140, 12324, 160020, 2424132, 41967540, 817374564, 17688328020, 421061260932, 10934334077940, 307610736087204, 9319558144624020, 302518807147502532, 10474617188712332340, 385347795973248950244, 15010362565222418008020, 617178205591321673884932

Offset: 0

Paul D. Hanna, Nov 30 2011

			E.g.f.: E(x) = 1 + 4*x + 20*x^2/2! + 132*x^3/3! + 1140*x^4/4! + 12324*x^5/5! + ...
O.g.f.: A(x) = 1 + 4*x + 20*x^2 + 132*x^3 + 1140*x^4 + 12324*x^5 + ...
where A(x) = 1 + 4*x/(1+3*x) + 2!*4^2*x^2/((1+3*x)*(1+6*x)) + 3!*4^3*x^3/((1+3*x)*(1+6*x)*(1+9*x)) + 4!*4^4*x^4/((1+3*x)*(1+6*x)*(1+9*x)*(1+12*x)) + ...

G. C. Greubel, Table of n, a(n) for n = 0..390

Cf. A201354, A000629, A123227.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( 3*Exp(3*x)/(4-Exp(3*x)) ))); // G. C. Greubel, Jun 09 2022

Table[Sum[(-3)^(n-k)*4^k*StirlingS2[n,k]*k!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 13 2013 *)
With[{nn=20},CoefficientList[Series[3 Exp[3x]/(4-Exp[3x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 03 2020 *)

{a(n)=n!*polcoeff(3*exp(3*x+x*O(x^n))/(4 - exp(3*x+x*O(x^n))), n)}

{a(n)=polcoeff(sum(m=0, n, 4^m*m!*x^m/prod(k=1, m, 1+3*k*x+x*O(x^n))), n)}

a(n)=sum(k=0, n, (-3)^(n-k)*4^k*stirling(n,k,2)*k!);

my(x='x+O('x^66)); Vec(serlaplace(3*exp(3*x)/(4-exp(3*x)))) \\ Joerg Arndt, May 06 2013

@CachedFunction
def BB(n, k, x):  # modified cardinal B-splines
    if n == 1: return 0 if (x < 0) or (x >= k) else 1
    return x*BB(n-1, k, x) + (n*k-x)*BB(n-1, k, x-k)
def EulerianPolynomial(n, k, x):
    if n == 0: return 1
    return add(BB(n+1, k, k*m+1)*x^m for m in (0..n))
[4^n*EulerianPolynomial(n,1,1/4) for n in (0..19)]  # Peter Luschny, May 04 2013

O.g.f.: A(x) = Sum_{n>=0} n! * 4^n*x^n / Product_{k=0..n} (1+3*k*x).
O.g.f.: A(x) = 1/(1 - 4*x/(1-x/(1 - 8*x/(1-2*x/(1 - 12*x/(1-3*x/(1 - 16*x/(1-4*x/(1 - 20*x/(1-5*x/(1 - ...))))))))))), a continued fraction.
a(n) = Sum_{k=0..n} (-3)^(n-k) * 4^k * Stirling2(n,k) * k!.
a(n) = Sum_{k=0..n} A123125(n,k)*4^k. - Philippe Deléham, Nov 30 2011
G.f.: 2/G(0), where G(k) = 1 + 1/(1 - 8*x*(k+1)/(8*x*(k+1) - 1 + 2*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
a(n) ~ n! * (3/log(4))^(n+1) . - Vaclav Kotesovec, Jun 13 2013
a(n) = 3^n*log(4)*Integral_{x = 0..oo} (ceiling(x))^n * 4^(-x) dx. - Peter Bala, Feb 06 2015
a(n) = 3^(n+1) * Sum_{k>=1} k^n / 4^k. - Ilya Gutkovskiy, Jun 28 2020

A345102 a(n) = 1 + 3 * Sum_{k=0..n-1} binomial(n,k) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 4, 37, 589, 13276, 386059, 13741057, 578451514, 28109736811, 1548565036789, 95365652263102, 6492034471389889, 484086370908869821, 39238367740327468444, 3435176518078688461297, 323029539924876486293089, 32472511993953383052630556, 3475005417300807667690138399

Offset: 0

Ilya Gutkovskiy, Jun 08 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..343

Cf. A006677, A011781, A052886, A201354, A345103.

a[n_] := a[n] = 1 + 3 Sum[Binomial[n, k] a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Exp[x]/Sqrt[7 - 6 Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Sum[Binomial[n, k] StirlingS2[k, j] 3^j (2 j - 1)!!, {j, 0, k}], {k, 0, n}], {n, 0, 17}]

N=20; x='x+O('x^N); Vec(serlaplace(exp(x)/sqrt(7-6*exp(x)))) \\ Seiichi Manyama, Oct 20 2021

E.g.f.: exp(x) / sqrt(7 - 6 * exp(x)).

A367490 Expansion of e.g.f. -x * log(4 - 3*exp(x)).

Original entry on oeis.org

0, 0, 6, 36, 336, 4380, 73080, 1481844, 35320992, 966875724, 29874822600, 1028081942052, 38985534525168, 1614899447153148, 72543518616692760, 3512306387815898580, 182320857226312198464, 10100520471366488756652, 594804877105749056467560

Offset: 0

Seiichi Manyama, Nov 19 2023

Cf. A052862, A367489.

Cf. A032183, A201354.

a(n) = n*sum(k=1, n-1, 3^k*(k-1)!*stirling(n-1, k, 2));

a(n) = n * Sum_{k=1..n-1} 3^k * (k-1)! * Stirling2(n-1,k).

A368322 Expansion of e.g.f. exp(2x) / (4 - 3exp(x)).

Original entry on oeis.org

1, 5, 37, 389, 5413, 94085, 1962277, 47746949, 1327769893, 41538664325, 1443908686117, 55210237509509, 2302968844974373, 104068337416767365, 5064468256286449957, 264065894676248072069, 14686540175450593986853, 867871886679723760867205

Offset: 0

Seiichi Manyama, Dec 21 2023

nonn

Cf. A032033, A201354, A368323, A368324.

b(n, t) = sum(k=0, n, t^k*k!*stirling(n, k, 2));
a(n, m=2, t=3) = my(u=1+1/t); u^m*b(n, t)-(1/t)*sum(j=0, m-1, u^j*(m-1-j)^n);

a(n) = 2^n + 3 * Sum_{k=1..n} binomial(n,k) * a(n-k).

a(n) = (16/9)*A032033(n) - (1/3)*(1 + (4/3)*0^n).

A368323 Expansion of e.g.f. exp(3x) / (4 - 3exp(x)).

Original entry on oeis.org

1, 6, 48, 516, 7212, 125436, 2616348, 63662556, 1770359772, 55384885596, 1925211581148, 73613650011996, 3070625126631132, 138757783222353756, 6752624341715261148, 352087859568330751836, 19582053567267458627292, 1157162515572965014445916

Offset: 0

Seiichi Manyama, Dec 21 2023

nonn

Cf. A032033, A201354, A368322, A368324.

With[{nn=20},CoefficientList[Series[Exp[3x]/(4-3Exp[x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 18 2025 *)

b(n, t) = sum(k=0, n, t^k*k!*stirling(n, k, 2));
a(n, m=3, t=3) = my(u=1+1/t); u^m*b(n, t)-(1/t)*sum(j=0, m-1, u^j*(m-1-j)^n);

a(n) = 3^n + 3 * Sum_{k=1..n} binomial(n,k) * a(n-k).

a(n) = (64/27)*A032033(n) - (1/3)*(2^n + 4/3 + (16/9)*0^n).

cp's OEIS Frontend

A201365 Expansion of e.g.f. exp(x) / (5 - 4*exp(x)).

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A278075 Coefficients of the signed Fubini polynomials in ascending order, F_n(x) = Sum_{k=0..n} (-1)^n*Stirling2(n,k)*k!*(-x)^k.

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A090355 G.f. satisfies A^4 = BINOMIAL(A)^3.

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A136728 E.g.f.: A(x) = (exp(x)/(4 - 3*exp(x)))^(1/4).

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A343709 a(n) = 1 + 3 * Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).

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A201355 Expansion of e.g.f.: 3*exp(3*x) / (4 - exp(3*x)).

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Magma

Mathematica

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PARI

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Sage

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A345102 a(n) = 1 + 3 * Sum_{k=0..n-1} binomial(n,k) * a(k) * a(n-k-1).

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A278075 Coefficients of the signed Fubini polynomials in ascending order, F_n(x) = Sum_{k=0..n} (-1)^nStirling2(n,k)k!*(-x)^k.

A201355 Expansion of e.g.f.: 3exp(3x) / (4 - exp(3*x)).

A368322 Expansion of e.g.f. exp(2x) / (4 - 3exp(x)).

A368323 Expansion of e.g.f. exp(3x) / (4 - 3exp(x)).