cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

1, 5, 45, 605, 10845, 243005, 6534045, 204972605, 7348546845, 296387331005, 13282361478045, 654762261324605, 35211177242722845, 2051349014835939005, 128701394409842982045, 8651475271312083756605, 620334325261670875138845, 47259638324026516284867005
Offset: 0

Views

Author

Paul D. Hanna, Nov 30 2011

Keywords

Examples

			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)) + ...

Links

Crossrefs

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

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!(Laplace( 1/(5*Exp(-x) -4) ))); // G. C. Greubel, Jun 08 2020
  • Maple
    seq(coeff(series(1/(5*exp(-x) - 4), x, n+1)*n!, x, n), n = 0..20); # G. C. Greubel, Jun 08 2020
  • Mathematica
    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 *)
  • PARI
    {a(n)=n!*polcoeff(exp(x+x*O(x^n))/(5 - 4*exp(x+x*O(x^n))), n)}
  • PARI
    {a(n)=polcoeff(sum(m=0, n, 5^m*m!*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)}
  • PARI
    {a(n)=sum(k=0, n, (-1)^(n-k)*5^k*stirling(n, k, 2)*k!)}
  • Sage
    [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

Formula

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

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

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

Views

Author

Paul D. Hanna, Nov 30 2011

Keywords

Examples

			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)) + ...

Links

Crossrefs

Cf. A201365, A201366, A201368.

Programs

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

Formula

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.: 4*exp(4*x) / (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

Views

Author

Paul D. Hanna, Nov 30 2011

Keywords

Examples

			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)) + ...

Crossrefs

Cf. A201365, A201366, A201367.

Programs

  • Mathematica
    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 *)
  • PARI
    {a(n)=n!*polcoeff(4*exp(4*x+x*O(x^n))/(5 - exp(4*x+x*O(x^n))), n)}
  • PARI
    {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)}
  • PARI
    {a(n)=sum(k=0, n, (-4)^(n-k)*5^k*stirling(n, k, 2)*k!)}
  • PARI
    my(x='x+O('x^66)); Vec(serlaplace(4*exp(4*x)/(5-exp(4*x)))) \\ Joerg Arndt, May 06 2013
  • Sage
    @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

Formula

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

A384435 Expansion of e.g.f. 2/(5 - 3*exp(2*x)).

Original entry on oeis.org

1, 3, 24, 282, 4416, 86448, 2030784, 55656912, 1743277056, 61427981568, 2405046994944, 103579443604992, 4866448609591296, 247692476576575488, 13576823521525653504, 797345878311609526272, 49948684871884896731136, 3324530341927517641310208, 234293439367907438337982464
Offset: 0

Views

Author

Seiichi Manyama, Jun 03 2025

Keywords

Links

Crossrefs

Cf. A094417, A326324, A382753.
Cf. A032033, A328182, A384522.
Cf. A201366.

Programs

  • PARI
    a(n) = (-2)^(n+1)*polylog(-n, 5/3)/5;

Formula

a(n) = (-2)^(n+1)/5 * Li_{-n}(5/3), where Li_{n}(x) is the polylogarithm function.
a(n) = 2^(n+1)/5 * Sum_{k>=0} k^n * (3/5)^k.
a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * k! * Stirling2(n,k).
a(n) = (3/5) * A201366(n) = (3/5) * Sum_{k=0..n} 5^k * (-2)^(n-k) * k! * Stirling2(n,k) for n > 0.
a(0) = 1; a(n) = 3 * Sum_{k=1..n} 2^(k-1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 3 * a(n-1) + 5 * Sum_{k=1..n-1} (-2)^(k-1) * binomial(n-1,k) * a(n-k).
Showing 1-4 of 4 results.

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