A201365 -id:A201365 - cp's OEIS frontend

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

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

A090357 G.f. satisfies A^5 = BINOMIAL(A)^4; also equals A090356^4.

Original entry on oeis.org

1, 4, 26, 244, 3131, 52600, 1111940, 28559320, 865622825, 30250881420, 1196941704454, 52860066623036, 2576115583371739, 137274420821505776, 7937914900025008984, 494941882189888642832, 33096552232229291234923

Offset: 0

Paul D. Hanna, Nov 26 2003

See comments in A090356.

Cf. A090356, A019538, A050353, A201365.

Cf. A090352, A090355, A090362.

nmax = 16; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x]^5 - A[x/(1 - x)]^4/(1 - x)^4 + 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^5+B^4);polcoeff(A,n,x))}

G.f.: A(x)^5 = A(x/(1-x))^4/(1-x)^4.
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)*4^k = A094417(n) = 4*A050353(n) for n >= 1.
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)*(-5)^k = A201365(n) = 5*A050353(n) for n >= 1.
A(x) = B(x)^4 and BINOMIAL(A(x)) = B(x)^5 where B(x) = 1 + x + 5*x^2 + 45*x^3 + 495*x^4 + ... is the o.g.f. for A090356. See also A019538. (End)
G.f.: Product_{k>=1} 1/(1 - k*x)^((1/5) * (4/5)^k). - Seiichi Manyama, May 26 2025
a(n) ~ (n-1)! / (5 * log(5/4)^(n+1)). - Vaclav Kotesovec, May 28 2025

A201367 E.g.f.: 3exp(3x) / (5 - 2exp(3x)).

Original entry on oeis.org

1, 5, 35, 345, 4515, 73905, 1451835, 33273945, 871529715, 25681042305, 840815302635, 30281769805545, 1189735610250915, 50638609760802705, 2321120945531697435, 113992686944812385145, 5971520591679167948115, 332369999588147180115105, 19587647624733292373756235

Offset: 0

Paul D. Hanna, Nov 30 2011

			E.g.f.: E(x) = 1 + 5*x + 35*x^2/2! + 345*x^3/3! + 4515*x^4/4! + 73905*x^5/5! + ...
O.g.f.: A(x) = 1 + 5*x + 35*x^2 + 345*x^3 + 4515*x^4 + 73905*x^5 + ...
where A(x) = 1 + 5*x/(1+3*x) + 2!*5^2*x^2/((1+3*x)*(1+6*x)) + 3!*5^3*x^3/((1+3*x)*(1+6*x)*(1+9*x)) + 4!*5^4*x^4/((1+3*x)*(1+6*x)*(1+9*x)*(1+12*x)) + ...

Robert Israel, Table of n, a(n) for n = 0..250

Cf. A201365, A201366, A201368.

S:= series(3*exp(3*x)/(5-2*exp(3*x)),x,51):
seq(coeff(S,x,n)*n!,n=0..50); # Robert Israel, Nov 18 2019

Table[Sum[(-3)^(n-k)*5^k*StirlingS2[n,k]*k!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 13 2013 *)
With[{nn=20},CoefficientList[Series[(3*Exp[3x])/(5-2*Exp[3x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 07 2024 *)

{a(n)=n!*polcoeff(3*exp(3*x+x*O(x^n))/(5 - 2*exp(3*x+x*O(x^n))), n)}

{a(n)=polcoeff(sum(m=0, n, 5^m*m!*x^m/prod(k=1, m, 1+3*k*x+x*O(x^n))), n)}

{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, (-3)^(n-k)*5^k*Stirling2(n, k)*k!)}

O.g.f.: A(x) = Sum_{n>=0} n! * 5^n*x^n / Product_{k=0..n} (1+3*k*x).
O.g.f.: A(x) = 1/(1 - 5*x/(1-2*x/(1 - 10*x/(1-4*x/(1 - 15*x/(1-6*x/(1 - 20*x/(1-8*x/(1 - 25*x/(1-10*x/(1 - ...))))))))))), a continued fraction.
a(n) = Sum_{k=0..n} (-3)^(n-k) * 5^k * Stirling2(n,k) * k!.
a(n) = Sum_{k=0..n} A123125(n,k)*5^k*2*(n-k). - Philippe Deléham, Nov 30 2011
a(n) ~ n! / (2*(log(5/2)/3)^(n+1)). - Vaclav Kotesovec, Jun 13 2013
a(n) = 3^n*log(5/2) * Integral_{x = 0..oo} (ceiling(x))^n * (5/2)^(-x) dx. - Peter Bala, Feb 06 2015

A201368 E.g.f.: 4exp(4x) / (5 - exp(4*x)).

Original entry on oeis.org

1, 5, 30, 230, 2280, 28280, 421680, 7336880, 145879680, 3263031680, 81097294080, 2217097729280, 66122900014080, 2136392343342080, 74335250629908480, 2771225281718343680, 110198981079416340480, 4655992415884353044480, 208291013498682750074880, 9835804726301090178990080

Offset: 0

Paul D. Hanna, Nov 30 2011

			E.g.f.: E(x) = 1 + 5*x + 30*x^2/2! + 230*x^3/3! + 2280*x^4/4! + 28280*x^5/5! + ...
O.g.f.: A(x) = 1 + 5*x + 30*x^2 + 230*x^3 + 2280*x^4 + 28280*x^5 + ...
where A(x) = 1 + 5*x/(1+4*x) + 2!*5^2*x^2/((1+4*x)*(1+8*x)) + 3!*5^3*x^3/((1+4*x)*(1+8*x)*(1+12*x)) + 4!*5^4*x^4/((1+4*x)*(1+8*x)*(1+12*x)*(1+16*x)) + ...

Cf. A201365, A201366, A201367.

Table[Sum[(-4)^(n-k)*5^k*StirlingS2[n,k]*k!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 13 2013 *)
With[{nn=20},CoefficientList[Series[4 Exp[4x]/(5-Exp[4x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 25 2024 *)

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

{a(n)=polcoeff(sum(m=0, n, 5^m*m!*x^m/prod(k=1, m, 1+4*k*x+x*O(x^n))), n)}

{a(n)=sum(k=0, n, (-4)^(n-k)*5^k*stirling(n, k, 2)*k!)}

my(x='x+O('x^66)); Vec(serlaplace(4*exp(4*x)/(5-exp(4*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))
[5^n*EulerianPolynomial(n, 1, 1/5) for n in (0..19)]   # Peter Luschny, May 04 2013

O.g.f.: A(x) = Sum_{n>=0} n! * 5^n*x^n / Product_{k=0..n} (1+4*k*x).
O.g.f.: A(x) = 1/(1 - 5*x/(1-x/(1 - 10*x/(1-2*x/(1 - 15*x/(1-3*x/(1 - 20*x/(1-4*x/(1 - 25*x/(1-5*x/(1 - ...))))))))))), a continued fraction.
a(n) = Sum_{k=0..n} (-4)^(n-k) * 5^k * Stirling2(n,k) * k!.
a(n) = Sum_{k=0..n} A123125(n,k)*5^k. - Philippe Deléham, Nov 30 2011
a(n) ~ n! * (4/log(5))^(n+1). - Vaclav Kotesovec, Jun 13 2013
a(n) = 4^n*log(5) * Integral_{x = 0..oo} (ceiling(x))^n * 5^(-x) dx. - Peter Bala, Feb 06 2015
a(n) = 4^(n+1) * Sum_{k>=1} k^n / 5^k. - Ilya Gutkovskiy, Jun 28 2020

A201366 E.g.f.: 2exp(2x) / (5 - 3exp(2x)).

Original entry on oeis.org

1, 5, 40, 470, 7360, 144080, 3384640, 92761520, 2905461760, 102379969280, 4008411658240, 172632406008320, 8110747682652160, 412820794294292480, 22628039202542755840, 1328909797186015877120, 83247808119808161218560, 5540883903212529402183680, 390489065613179063896637440

Offset: 0

Paul D. Hanna, Nov 30 2011

			E.g.f.: E(x) = 1 + 5*x + 40*x^2/2! + 470*x^3/3! + 7360*x^4/4! + 144080*x^5/5! + ...
O.g.f.: A(x) = 1 + 5*x + 40*x^2 + 470*x^3 + 7360*x^4 + 144080*x^5 + ...
where A(x) = 1 + 5*x/(1+2*x) + 2!*5^2*x^2/((1+2*x)*(1+4*x)) + 3!*5^3*x^3/((1+2*x)*(1+4*x)*(1+6*x)) + 4!*5^4*x^4/((1+2*x)*(1+4*x)*(1+6*x)*(1+8*x)) + ...

Cf. A201365, A201367, A201368.

Table[Sum[(-2)^(n-k)*5^k*StirlingS2[n,k]*k!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 13 2013 *)
With[{nn=20},CoefficientList[Series[(2Exp[2x])/(5-3Exp[2x]),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Dec 29 2018 *)

{a(n)=n!*polcoeff(2*exp(2*x+x*O(x^n))/(5 - 3*exp(2*x+x*O(x^n))), n)}

{a(n)=polcoeff(sum(m=0, n, 5^m*m!*x^m/prod(k=1, m, 1+2*k*x+x*O(x^n))), n)}

{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)*5^k*Stirling2(n, k)*k!)}

O.g.f.: A(x) = Sum_{n>=0} n! * 5^n*x^n / Product_{k=0..n} (1+2*k*x).
O.g.f.: A(x) = 1/(1 - 5*x/(1-3*x/(1 - 10*x/(1-6*x/(1 - 15*x/(1-9*x/(1 - 20*x/(1-12*x/(1 - 25*x/(1-15*x/(1 - ...))))))))))), a continued fraction.
a(n) = Sum_{k=0..n} (-2)^(n-k) * 5^k * Stirling2(n,k) * k!.
a(n) = Sum_{k=0..n} A123125(n,k)*5^k*3^(n-k). - Philippe Deléham, Nov 30 2011
a(n) ~ n! / (3*(log(5/3)/2)^(n+1)). - Vaclav Kotesovec, Jun 13 2013
a(n) = 2^n*log(5/3) * Integral_{x = 0..oo} (ceiling(x))^n * (5/3)^(-x) dx. - Peter Bala, Feb 06 2015

A343710 a(n) = 1 + 4 * Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).

Original entry on oeis.org

1, 5, 45, 609, 11009, 248837, 6749629, 213596401, 7725031521, 314310704101, 14209394894765, 706617979262049, 38333841625642785, 2252901018519028901, 142589176837851349757, 9669282207517755852721, 699408060608904410296897, 53752166013267632536864581, 4374061543586452325644329133

Offset: 0

Ilya Gutkovskiy, Apr 26 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..359

Cf. A201365, A291979, A343707, A343709.

a[n_] := a[n] = 1 + 4 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 + 4 Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!

N=20; x='x+O('x^N); Vec(serlaplace(exp(x)/(1+4*log(1-x)))) \\ Seiichi Manyama, Oct 20 2021

E.g.f.: exp(x) / (1 + 4 * log(1 - x)).

A345103 a(n) = 1 + 4 * Sum_{k=0..n-1} binomial(n,k) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 5, 61, 1277, 37741, 1437725, 67013101, 3693540317, 234974905261, 16945434018845, 1366008048556141, 121721015465713757, 11880107754103150381, 1260413749895624939165, 144427420001275864755181, 17776090894283922227621597, 2338833689096321086977341101, 327585830473259220341296486685

Offset: 0

Ilya Gutkovskiy, Jun 08 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..331

Cf. A006677, A052886, A144828, A201365, A345102.

a[n_] := a[n] = 1 + 4 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[9 - 8 Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Sum[Binomial[n, k] StirlingS2[k, j] 4^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(9-8*exp(x)))) \\ Seiichi Manyama, Oct 20 2021

E.g.f.: exp(x) / sqrt(9 - 8 * exp(x)).

A367374 Expansion of the e.g.f. (exp(x) / (5 - 4*exp(x)))^(2/5).

Original entry on oeis.org

1, 2, 12, 128, 1944, 38264, 924936, 26507672, 878565000, 33058419032, 1392125985864, 64864749910424, 3313075222410504, 184071465908101592, 11051901784679926728, 713107430713993422872, 49208366812318404125832, 3616200105869781814285400

Offset: 0

Seiichi Manyama, Nov 15 2023

nonn

Cf. A136729, A201365, A367375, A367376.

Cf. A365568.

a(n) = sum(k=0, n, (-1)^(n-k)*prod(j=0, k-1, 5*j+2)*stirling(n, k, 2));

a(n) = Sum_{k=0..n} (-1)^(n-k) * (Product_{j=0..k-1} (5*j+2)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^k * (3*k/n - 5) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 2*a(n-1) + 4*Sum_{k=1..n-1} binomial(n-1,k) * a(n-k).

A367375 Expansion of the e.g.f. (exp(x) / (5 - 4*exp(x)))^(3/5).

Original entry on oeis.org

1, 3, 21, 243, 3909, 80451, 2016885, 59610771, 2029183653, 78173046243, 3362038875093, 159665003673651, 8298290454862341, 468484406336978307, 28548397948780827957, 1867633303272817927635, 130551162799758211802469, 9710901131124428156535075

Offset: 0

Seiichi Manyama, Nov 15 2023

nonn

Cf. A136729, A201365, A367374, A367376.

Cf. A365569.

a(n) = sum(k=0, n, (-1)^(n-k)*prod(j=0, k-1, 5*j+3)*stirling(n, k, 2));

a(n) = Sum_{k=0..n} (-1)^(n-k) * (Product_{j=0..k-1} (5*j+3)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^k * (2*k/n - 5) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 3*a(n-1) + 4*Sum_{k=1..n-1} binomial(n-1,k) * a(n-k).

A367376 Expansion of the e.g.f. (exp(x) / (5 - 4*exp(x)))^(4/5).

Original entry on oeis.org

1, 4, 32, 400, 6800, 146128, 3795728, 115616848, 4040024720, 159282704848, 6993908053520, 338443123424080, 17894609985867152, 1026351961130219728, 63466858180767590672, 4209071260503851502160, 298006515851074633361552, 22434758711582422326267856

Offset: 0

Seiichi Manyama, Nov 15 2023

nonn

Cf. A136729, A201365, A367374, A367375.

Cf. A365570, A365587, A365603.

a(n) = sum(k=0, n, (-1)^(n-k)*prod(j=0, k-1, 5*j+4)*stirling(n, k, 2));

a(n) = Sum_{k=0..n} (-1)^(n-k) * (Product_{j=0..k-1} (5*j+4)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^k * (k/n - 5) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 4*a(n-1) + 4*Sum_{k=1..n-1} binomial(n-1,k) * a(n-k).

cp's OEIS Frontend

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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A090357 G.f. satisfies A^5 = BINOMIAL(A)^4; also equals A090356^4.

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A201367 E.g.f.: 3*exp(3*x) / (5 - 2*exp(3*x)).

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A201368 E.g.f.: 4*exp(4*x) / (5 - exp(4*x)).

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A201366 E.g.f.: 2*exp(2*x) / (5 - 3*exp(2*x)).

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A343710 a(n) = 1 + 4 * Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).

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A345103 a(n) = 1 + 4 * 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.

A201367 E.g.f.: 3exp(3x) / (5 - 2exp(3x)).

A201368 E.g.f.: 4exp(4x) / (5 - exp(4*x)).

A201366 E.g.f.: 2exp(2x) / (5 - 3exp(2x)).