A201339 -id:A201339 - 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

A090352 G.f. satisfies A^3 = BINOMIAL(A)^2, where A = A090351^2.

Original entry on oeis.org

1, 2, 7, 36, 255, 2370, 27713, 393352, 6582068, 126888632, 2767912036, 67362737168, 1808596304964, 53083358012760, 1690443996202428, 58039582729688320, 2136931230333535178, 83981145793974066484

Offset: 0

Paul D. Hanna, Nov 26 2003

See comments in A090351.

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

Cf. A004123, A019538, A050351, A084785, A090351, A201339.

Cf. A090355, A090357, A090362.

m:=40;
f:= func< n, x | Exp((&+[(&+[2^j*Factorial(j)*StirlingSecond(k, j)*x^k/k: j in [1..k]]): k in [1..n+2]])) >;
R:=PowerSeriesRing(Rationals(), m+1);  // A090352
Coefficients(R!( f(m, x) )); // G. C. Greubel, Jul 07 2023

nmax = 17; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x]^3 - A[x/(1 - x)]^2/(1 - x)^2 + 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^3+B^2); polcoeff(A,n,x))}

m=50
def f(n, x): return exp(sum(sum(2^j*factorial(j)*stirling_number2(k, j)*x^k/k for j in range(1, k+1)) for k in range(1, n+2)))
def A090352_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(m, x) ).list()
A090352_list(m-9) # G. C. Greubel, Jul 07 2023

G.f. satisfies: A(x)^3 = A(x/(1-x))^2/(1-x)^2.
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)*2^k = A004123(n+1) = 2*A050351(n) for n >= 1. Cf. A084785.
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)*(-3)^k = A201339(n) = 3*A050351(n) for n >= 1.
A(x) = B(x)^2 and BINOMIAL(A(x)) = B(x)^3 where B(x) = 1 + x + 3*x^2 + 15*x^3 + 108*x^4 + ... is the o.g.f. for A090351. See also A019538. (End)
G.f.: Product_{k>=1} 1/(1 - k*x)^((1/3) * (2/3)^k). - Seiichi Manyama, May 26 2025
a(n) ~ (n-1)! / (3 * log(3/2)^(n+1)). - Vaclav Kotesovec, May 28 2025

A123227 Expansion of e.g.f.: 2exp(2x) / (3 - exp(2*x)).

Original entry on oeis.org

1, 3, 12, 66, 480, 4368, 47712, 608016, 8855040, 145083648, 2641216512, 52891055616, 1155444326400, 27344999497728, 696933753434112, 19031293222127616, 554336947975618560, 17155693983744196608, 562168282464340672512, 19444889661250162262016

Offset: 0

Philippe Deléham, Oct 06 2006

G. C. Greubel, Table of n, a(n) for n = 0..200
Paul Barry, Eulerian-Dowling Polynomials as Moments, Using Riordan Arrays, arXiv:1702.04007 [math.CO], 2017.
Eric Weisstein's MathWorld, Polylogarithm.
Eric Weisstein's MathWorld, Lerch Transcendent.

Cf. A000629, A201339, A122704, A009362, A123125, A028246.

a := n -> 2^(n+1)*polylog(-n, 1/3):
seq(round(evalf(a(n),32)), n=0..19); # Peter Luschny, Nov 03 2015
seq(expand(2^(n+1)*polylog(-n,1/3)), n=0..100); # Robert Israel, Nov 03 2015

CoefficientList[Series[2*Exp[2*x]/(3-Exp[2*x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 24 2013 *)
Round@Table[(-1)^(n+1) (LerchPhi[Sqrt[3], -n, 0] + LerchPhi[-Sqrt[3], -n, 0]), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 31 2015 *)

{a(n)=n!*polcoeff(2*exp(2*x+x*O(x^n))/(3 - exp(2*x+x*O(x^n))), n)} /* Paul D. Hanna */

{a(n)=polcoeff(sum(m=0, n, 3^m*m!*x^m/prod(k=1, m, 1+2*k*x+x*O(x^n))), n)} /* Paul D. Hanna */

{Stirling2(n, k)=if(k<0||k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}
{a(n)=sum(k=0, n, (-2)^(n-k)*3^k*Stirling2(n, k)*k!)} /* Paul D. Hanna */

my(x='x+O('x^20)); Vec(serlaplace(2*exp(2*x)/(3-exp(2*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))
def A123227(n) : return 3^n*EulerianPolynomial(n, 1, 1/3)
[A123227(n) for n in (0..18)]  # Peter Luschny, May 04 2013

a(n) = abs(A009362(n+1)).
a(n-1) = Sum_{k=1..n} 2^(n-k)*A028246(n,k), n>=1.
a(n) = Sum_{k=0..n} 3^k*A123125(n,k).
From Paul D. Hanna, Nov 30 2011: (Start)
a(n) = 3*A122704(n) for n>0.
a(n) = Sum_{k=0..n} (-2)^(n-k) * 3^k * Stirling2(n,k) * k!.
O.g.f.: Sum_{n>=0} 3^n * n!*x^n / Product_{k=0..n} (1+2*k*x).
O.g.f.: 1/(1 - 3*x/(1-x/(1 - 6*x/(1-2*x/(1 - 9*x/(1-3*x/(1 - 12*x/(1-4*x/(1 - 15*x/(1-5*x/(1 - ...))))))))))), a continued fraction.
(End)
a(n) ~ n! * (2/log(3))^(n+1). - Vaclav Kotesovec, Jun 24 2013
a(n) = 2^n*log(3)*Integral_{x = 0..oo} (ceiling(x))^n * 3^(-x) dx. - Peter Bala, Feb 06 2015
a(n) = (-1)^(n+1)*(LerchPhi(sqrt(3), -n, 0) + LerchPhi(-sqrt(3), -n, 0)) = (-1)^(n+1)*(Li_{-n}(sqrt(3)) + Li_{-n}(-sqrt(3))) - 2*0^n, where Li_n(x) is the polylogarithm. - Vladimir Reshetnikov, Oct 31 2015
a(n) = 2^(n+1)*Li_{-n}(1/3). - Peter Luschny, Nov 03 2015
a(0) = 1; a(n) = 2 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k) * a(n-k-1). - Ilya Gutkovskiy, Jul 05 2020

Name changed and a(8) corrected by Paul D. Hanna, Nov 30 2011

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

A343707 a(n) = 1 + 2 * Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).

Original entry on oeis.org

1, 3, 15, 113, 1145, 14539, 221663, 3943281, 80173345, 1833831619, 46606646175, 1302954958689, 39737420405753, 1312901360002283, 46714233470065999, 1780859204826798401, 72416689888874547969, 3128792006916853876291, 143132514626658326870767, 6911638338982428907738641

Offset: 0

Ilya Gutkovskiy, Apr 26 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..389

Cf. A088500, A201339, A291979, A343709, A343710.

a[n_] := a[n] = 1 + 2 Sum[Binomial[n, k] (n - k - 1)! a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 19}]
nmax = 19; CoefficientList[Series[Exp[x]/(1 + 2 Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!

N=20; x='x+O('x^N); Vec(serlaplace(exp(x)/(1+2*log(1-x)))) \\ Seiichi Manyama, Oct 20 2021

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

a(n) = Sum_{k=0..n} binomial(n,k) * A088500(k).

A136727 E.g.f.: A(x) = (exp(x)/(3 - 2*exp(x)))^(1/3).

Original entry on oeis.org

1, 1, 3, 17, 139, 1481, 19443, 303297, 5480219, 112549881, 2589274883, 65957355377, 1842897053099, 56038776055081, 1842278768795923, 65109900167188257, 2461735422517374779, 99148196540813749081

Offset: 0

Paul D. Hanna, Jan 24 2008

nonn

G.f. of variant A014307 is B(x) = sqrt(exp(x)/(2-exp(x))), which satisfies: B(x) = 1 + integral(B(x)^3*exp(-x)).

			E.g.f.: A(x) = 1 + x + 3/2*x^2 + 17/6*x^3 + 139/24*x^4 + 1481/120*x^5 +...

Harvey P. Dale, Table of n, a(n) for n = 0..350

Cf. A201339, variants: A014307, A136728, A136729.

With[{nn=20},CoefficientList[Series[(Exp[x]/(3-2Exp[x]))^(1/3),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 26 2013 *)

{a(n) = n!*polcoeff((exp(x +x*O(x^n))/(3-2*exp(x +x*O(x^n))))^(1/3),n)}
for(n=0,25,print1(a(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)}
for(n=0,25,print1(a(n),", "))

E.g.f. A(x) satisfies: A(x) = 1 + integral( A(x)^4 * exp(-x) ).
O.g.f.: 1/(1 - x/(1-2*x/(1 - 4*x/(1-4*x/(1 - 7*x/(1-6*x/(1 - 10*x/(1-8*x/(1 - 13*x/(1-10*x/(1 - ...))))))))))), a continued fraction.
G.f.: 1/G(0) where G(k) = 1 - x*(3*k+1)/( 1 - 2*x*(k+1)/G(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 23 2013
a(n) ~ n! * sqrt(3)*2^(2/3)*Gamma(2/3)/(4*Pi*n^(2/3)*(log(3/2))^(n+1/3)). - Vaclav Kotesovec, Jun 25 2013
a(n) = 1 + 2 * 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} (3*j+1)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^k * (2*k/n - 3) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = a(n-1) + 2*Sum_{k=1..n-1} binomial(n-1,k) * a(n-k). (End)

A367489 Expansion of e.g.f. -x * log(3 - 2*exp(x)).

Original entry on oeis.org

0, 0, 4, 18, 120, 1110, 13140, 189042, 3197040, 62093358, 1361253900, 33236925546, 894243758760, 26281928034726, 837663638344260, 28775491618091490, 1059805146165293280, 41657455054069680414, 1740535210734651716220, 77029901631623181859674

Offset: 0

Seiichi Manyama, Nov 19 2023

Cf. A052862, A367490.

Cf. A027882, A201339.

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

a(n) = n * Sum_{k=1..n-1} 2^k * (k-1)! * Stirling2(n-1,k).

A368319 Expansion of e.g.f. exp(2x) / (3 - 2exp(x)).

Original entry on oeis.org

1, 4, 22, 166, 1642, 20254, 299722, 5174446, 102094042, 2266154014, 55890234922, 1516265078926, 44874837768442, 1438774580904574, 49678366226498122, 1837828899444250606, 72522300447277154842, 3040654011774599283934, 134985159308312666889322

Offset: 0

Seiichi Manyama, Dec 21 2023

nonn

Cf. A004123, A201339, A368320, A368321.

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

a(n) = (9/4)*A004123(n+1) - (1/2)*(1 + (3/2)*0^n).

A368320 Expansion of e.g.f. exp(3x) / (3 - 2exp(x)).

Original entry on oeis.org

1, 5, 31, 245, 2455, 30365, 449551, 7761605, 153140935, 3399230765, 83835351871, 2274397617365, 67312256650615, 2158161871352765, 74517549339738991, 2756743349166359525, 108783450670915699495, 4560981017661898860365, 202477738962469000202911

Offset: 0

Seiichi Manyama, Dec 21 2023

nonn

Cf. A004123, A201339, A368319, A368321.

b(n, t) = sum(k=0, n, t^k*k!*stirling(n, k, 2));
a(n, m=3, t=2) = 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 + 2 * Sum_{k=1..n} binomial(n,k) * a(n-k).

a(n) = (27/8)*A004123(n+1) - (1/2)*(2^n + 3/2 + (9/4)*0^n).

A368321 Expansion of e.g.f. exp(4x) / (3 - 2exp(x)).

Original entry on oeis.org

1, 6, 42, 354, 3642, 45426, 673962, 11641314, 229708122, 5098836306, 125752998282, 3411596337474, 100968384710202, 3237242806231986, 111776324007217002, 4135115023742364834, 163175176006352025882, 6841471526492783720466, 303716608443703306594122

Offset: 0

Seiichi Manyama, Dec 21 2023

nonn

Cf. A004123, A201339, A368319, A368320.

b(n, t) = sum(k=0, n, t^k*k!*stirling(n, k, 2));
a(n, m=4, t=2) = 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) = 4^n + 2 * Sum_{k=1..n} binomial(n,k) * a(n-k).

a(n) = (81/16)*A004123(n+1) - (1/2)*(3^n + (3/2)*2^n + 9/4 + (27/8)*0^n).

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

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A090352 G.f. satisfies A^3 = BINOMIAL(A)^2, where A = A090351^2.

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A123227 Expansion of e.g.f.: 2*exp(2*x) / (3 - exp(2*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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Julia

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A343707 a(n) = 1 + 2 * Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).

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A136727 E.g.f.: A(x) = (exp(x)/(3 - 2*exp(x)))^(1/3).

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A123227 Expansion of e.g.f.: 2exp(2x) / (3 - exp(2*x)).

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

A368319 Expansion of e.g.f. exp(2x) / (3 - 2exp(x)).

A368320 Expansion of e.g.f. exp(3x) / (3 - 2exp(x)).

A368321 Expansion of e.g.f. exp(4x) / (3 - 2exp(x)).