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 1-3 of 3 results.

A052802 E.g.f. satisfies A(x) = 1/(1 + log(1 - x*A(x))).

Original entry on oeis.org

1, 1, 5, 47, 660, 12414, 293552, 8374806, 280064600, 10747277832, 465597887592, 22479948822792, 1197060450322800, 69699159437088960, 4405397142701855232, 300408348609092268144, 21983809533066553697280
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Comments

Previous name was: A simple grammar.

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 47*x^3/3! + 660*x^4/4! +... [_Paul D. Hanna_, Aug 28 2008]

Links

Crossrefs

Cf. A052819. [From Paul D. Hanna, Aug 28 2008]
Cf. A052894, A198860.

Programs

  • Maple
    spec := [S,{C=Cycle(B),S=Sequence(C),B=Prod(S,Z)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    CoefficientList[1/x*InverseSeries[Series[x + x*Log[1-x], {x, 0, 20}], x],x] * Range[0, 19]! (* Vaclav Kotesovec, Jan 08 2014 *)
  • Maxima
    a(n):=sum(binomial(n+k,n)*k!*sum((-1)^(n+j)/(k-j)!*sum(stirling1(n,j-i)/i!,i,0,j),j,0,k),k,0,n)/(n+1); /* Vladimir Kruchinin, May 09 2013 */
  • PARI
    a(n)=n!*polcoeff((1/x)*serreverse(x+x*log(1-x +x*O(x^n))),n) \\ Paul D. Hanna, Aug 28 2008

Formula

From Paul D. Hanna, Aug 28 2008: (Start)
E.g.f. satisfies: A(x*(1 + log(1-x))) = 1/(1 + log(1-x)).
E.g.f. satisfies: A(x) = 1/(1 + log(1 - x*A(x))).
E.g.f.: A(x) = (1/x)*Series_Reversion[x + x*log(1-x)]. (End)
a(n)=sum(k=0..n, binomial(n+k,n)*k!*sum(j=0..k, (-1)^(n+j)/(k-j)!*sum(i=0..j, stirling1(n,j-i)/i!)))/(n+1); [Vladimir Kruchinin, May 09 2013]
a(n) ~ n^(n-1) * c^n / (sqrt(1+c) * exp(n) * (c-1)^(2*n+1)), where c = LambertW(exp(2)) = 1.5571455989976114... - Vaclav Kotesovec, Jan 08 2014
a(n) = (1/(n+1)!) * Sum_{k=0..n} (n+k)! * |Stirling1(n,k)|. - Seiichi Manyama, Nov 06 2023

Extensions

New name using e.g.f., Vaclav Kotesovec, Jan 08 2014

A227457 E.g.f. equals the series reversion of x - x*log(1+x).

Original entry on oeis.org

1, 2, 9, 68, 720, 9804, 163184, 3210192, 72870120, 1874721360, 53905894152, 1713195438624, 59633476003920, 2256257009704320, 92196226214092800, 4046446853549201664, 189845257963376620800, 9481546020840245199360, 502242773970728703225600, 28124368575613839072714240
Offset: 1

Views

Author

Paul D. Hanna, Jul 12 2013

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 9*x^3/3! + 68*x^4/4! + 720*x^5/5! +...
where A(x) = x/(1 - log(1+A(x))).
The e.g.f. satisfies:
(3) A(x) = x + x*log(1+x) + d/dx x^2*log(1+x)^2/2! + d^2/dx^2 x^3*log(1+x)^3/3! + d^3/dx^3 x^4*log(1+x)^4/4! +...
(4) log(A(x)/x) = log(1+x) + d/dx x*log(1+x)^2/2! + d^2/dx^2 x^2*log(1+x)^3/3! + d^3/dx^3 x^3*log(1+x)^4/4! +...

Crossrefs

Cf. A052819, A030178.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[x - x*Log[1+x],{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 10 2014 *)
  • PARI
    {a(n)=n!*polcoeff(serreverse(x-x*log(1+x +x*O(x^n))), n)}
for(n=1,25,print1(a(n),", "))
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^m*log(1+x+x*O(x^n))^m/m!)); n!*polcoeff(A, n)}
for(n=1,25,print1(a(n),", "))
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*log(1+x+x*O(x^n))^m/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1,25,print1(a(n),", "))
  • PARI
    {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{a(n)=sum(k=0,n-1,k!*Stirling1(n-1,k)*binomial(n+k-1,n-1))}
for(n=1,25,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n-1} k! * Stirling1(n-1,k) * binomial(n+k-1,n-1). [From a formula in A052819 due to Vladimir Kruchinin]
E.g.f. A(x) satisfies:
(1) A(x - x*log(1+x)) = x.
(2) A(x) = x/(1 - log(1+A(x))).
(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n * log(1+x)^n / n!.
(4) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1) * log(1+x)^n / n! ).
a(n) ~ n^(n-1) * (1-c) / (c*sqrt(1+c) * exp(n) * (c-2+1/c)^n), where c = LambertW(1) = 0.5671432904... (see A030178). - Vaclav Kotesovec, Jan 10 2014

A368583 Table read by rows: T(n, k) = A124320(n + 1, k) * A132393(n, k).

Original entry on oeis.org

1, 0, 2, 0, 3, 12, 0, 8, 60, 120, 0, 30, 330, 1260, 1680, 0, 144, 2100, 11760, 30240, 30240, 0, 840, 15344, 113400, 428400, 831600, 665280, 0, 5760, 127008, 1169280, 5821200, 16632000, 25945920, 17297280, 0, 45360, 1176120, 13000680, 80415720, 302702400, 696215520, 908107200, 518918400
Offset: 0

Views

Author

Peter Luschny, Jan 10 2024

Keywords

Examples

			Triangle starts:
  [0] [1]
  [1] [0,   2]
  [2] [0,   3,   12]
  [3] [0,   8,   60,     120]
  [4] [0,  30,   330,   1260,   1680]
  [5] [0, 144,  2100,  11760,  30240,  30240]
  [6] [0, 840, 15344, 113400, 428400, 831600, 665280]

Crossrefs

Cf. A124320 (rising factorial), A132393 (unsigned Stirling1), A001813 (main diagonal), A052819 (row sums), A227457 (alternating row sums), A368584.

Programs

  • SageMath
    def Trow(n): return [rising_factorial(n+1, k)*stirling_number1(n, k)
                     for k in range(n+1)]
for n in range(7): print(Trow(n))
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.