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.

A087962 Satisfies Sum_{n>=0} a(n)*x^n/n! = log(f(x)) = series reversion of x*f(x), where f(x*f(x)) = exp(x) and f(x) = Sum_{n>=0} A087961(n)*x^n/n!.

Original entry on oeis.org

0, 1, -2, 15, -220, 5025, -159606, 6593041, -338977416, 21032339985, -1539275365450, 130569297615801, -12660181105282668, 1387510663815243721, -170295099173001030606, 23224872340978381412865, -3496270002640563444940816, 577651124287028261031912609
Offset: 0

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Author

Paul D. Hanna, Sep 18 2003

Keywords

Comments

This is the series reversion of xf(x) where f(xf(x))=exp(x), exp(xf(x))=f(xf(x)*exp(x)), f(log(x)*f(log(x)))=x and f(x)=sum(n>=0, A087961(n)*x^n/n!). Are these series convergent anywhere besides at x=0?

Examples

			f(x) = 1 +1x -1x^2/2! +10x^3/3! -159x^4/4! +3816x^5/5! -125375x^6/6! +-...
where f(xf(x)) = exp(x).

Links

Crossrefs

Cf. A087961.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(n=0, 1/k, add(k*
      b(j-1, j)*j*b(n-j, k)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> -b(n-1, n)*n*(-1)^n:
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 21 2019
  • Mathematica
    b[n_, k_] := b[n, k] = If[n == 0, 1/k, Sum[k*
     b[j-1, j]*j*b[n-j, k]*Binomial[n-1, j-1], {j, 1, n}]];
a[n_] := -b[n-1, n]*n*(-1)^n;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 21 2022, after Alois P. Heinz *)

A144681 E.g.f. satisfies: A(x/A(x)) = exp(x).

Original entry on oeis.org

1, 1, 3, 22, 305, 6656, 204337, 8226436, 414585425, 25315924960, 1828704716801, 153433983789164, 14739472821255481, 1602471473448455104, 195300935112810494801, 26470100501608768436716
Offset: 0

Views

Author

Paul D. Hanna, Sep 19 2008

Keywords

Examples

			E.g.f. A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 305*x^4/4! +...
A(x/A(x)) = 1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! +...
1/A(x) = 1 + x - x^2/2! + 10*x^3/3! - 159*x^4/4! + 3816*x^5/5! -+...
A(log(A(x))) = 1 + x + 5*x^2/2! + 55*x^3/3! + 1005*x^4/4! + 26601*x^5/5! +...
ILLUSTRATE FORMULA a(n+1) = [x^n/n!] exp(x)*A(x)^(n+1) as follows.
Form a table of coefficients of x^k/k! in exp(x)*A(x)^n for n>=1, k>=0:
exp(x)*A(x)^1: [(1), 2, 6, 35, 416, 8437, 249340, ...];
exp(x)*A(x)^2: [1,(3), 13, 93, 1145, 22593, 645741, ...];
exp(x)*A(x)^3: [1, 4,(22), 181, 2320, 45199, 1257364, ...];
exp(x)*A(x)^4: [1, 5, 33,(305), 4097, 79825, 2177329, ...];
exp(x)*A(x)^5: [1, 6, 46, 471,(6656), 131001, 3529836, ...];
exp(x)*A(x)^6: [1, 7, 61, 685, 10201,(204337), 5477005, ...];
exp(x)*A(x)^7: [1, 8, 78, 953, 14960, 306643,(8226436), ...]; ...
then the terms along the main diagonal form this sequence shift left.

Links

Crossrefs

Cf. A144682, A144683, A144684, A087961, A140049.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(n=0,n,A=exp(serreverse(x/A)));n!*polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x+sum(k=2,n-1,a(k)*x^k/k!)+x*O(x^n));if(n==0,1,(n-1)!*polcoeff(exp(x+x*O(x^n))*A^n,n-1))}

Formula

E.g.f. satisfies: A(x) = exp( x*A(log A(x)) ).
E.g.f. satisfies: a(n+1) = [x^n/n!] exp(x)*A(x)^(n+1).
E.g.f: A(x) = G(2x)^(1/2) where G(x/G(x)^2) = exp(x) and G(x) is the e.g.f. of A144682.
E.g.f: A(x) = G(3x)^(1/3) where G(x/G(x)^3) = exp(x) and G(x) is the e.g.f. of A144683.
E.g.f: A(x) = G(4x)^(1/4) where G(x/G(x)^4) = exp(x) and G(x) is the e.g.f. of A144684.
E.g.f: A(x) = 1/G(-x) where G(x*G(x)) = exp(x) and G(x) is the e.g.f. of A087961.
E.g.f. A(log(A(x))) = log(A(x))/x = G(x) is the e.g.f of A140049 where G(x) satisfies G(x*exp(-x*G(x))) = exp(x*G(x)).

A140054 E.g.f. A(x) satisfies: A( x*exp(-A(x)) ) = x.

Original entry on oeis.org

1, 2, 15, 220, 5025, 159606, 6593041, 338977416, 21032339985, 1539275365450, 130569297615801, 12660181105282668, 1387510663815243721, 170295099173001030606, 23224872340978381412865, 3496270002640563444940816, 577651124287028261031912609, 104221856744783499072505465746
Offset: 1

Views

Author

Paul D. Hanna, May 03 2008

Keywords

Comments

Unsigned version of A087962.
Not the same as A178533.

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 15*x^3/3! + 220*x^4/4! + 5025*x^5/5! +...
Related expansions are:
exp(-A(x)) = 1 - x - x^2/2! - 10*x^3/3! - 159*x^4/4! - 3816*x^5/5! -...
A(A(x)) = x + 4*x^2/2! + 42*x^3/3! + 764*x^4/4! + 20400*x^5/5! +...
A(A(A(x))) = x + 6*x^2/2! + 81*x^3/3! + 1776*x^4/4! + 55125*x^5/5! +...
A(A(A(A(x)))) = x + 8*x^2/2! + 132*x^3/3! + 3400*x^4/4! + 121080*x^5/5! +...
Iterations of A(x) obey the relation illustrated by:
A(x) = x*exp( A(A(x)) );
A(A(x)) = x*exp( A(A(x)) + A(A(A(x))) );
A(A(A(x))) = x*exp( A(A(x)) + A(A(A(x))) + A(A(A(A(x)))) ).
...

Links

Crossrefs

Cf. A087962 (A(-x)), A087961 (exp(-A(-x))), A140055 (A(A(x))).
Cf. A178533.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(n=0, 1/k, add(k*
      b(j-1, j)*j*b(n-j, k)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n-1, n)*n:
seq(a(n), n=1..20);  # Alois P. Heinz, Aug 21 2019
  • Mathematica
    b[n_, k_] := b[n, k] = If[n == 0, 1/k, Sum[k*b[j - 1, j]*j*b[n - j, k]* Binomial[n - 1, j - 1], {j, 1, n}]];
a[n_] := b[n - 1, n]*n;
a /@ Range[1, 20] (* Jean-François Alcover, Oct 03 2019, after Alois P. Heinz *)
  • Maxima
    A(n,m):=if n=m then 1 else m/n*sum(A(n-m,k)*n^k/k!,k,1,n-m);
makelist(n!*A(n,1),n,1,10); [Vladimir Kruchinin, May 06 2012]
  • PARI
    {a(n)=local(A=x);for(i=0,n,A=serreverse(x*exp(-A+x*O(x^n))));n!*polcoeff(A,n)}
for(n=1,20,print1(a(n),", "))
  • PARI
    {a(n)=local(A=x);for(i=0,n,A=x*exp(subst(A,x,A+x*O(x^n))));n!*polcoeff(A,n)}
for(n=1,20,print1(a(n),", "))

Formula

E.g.f. A(x) satisfies:
(1) A(x) = x*exp( A(A(x)) ).
(2) A(x) = x*exp( A(x)*exp( A(A(x))*exp( A(A(A(x)))*exp( ...)))) (infinite exponential tower).
(3) Let A_{n}(x) denote n-th iteration of e.g.f. A(x) with A_0(x)=x,
then
(3.a) A_{n+1}(x) = A( A_{n}(x) ) = A_{n}(x) * exp( A_{n+2}(x) );
(3.b) A_{n}(x) = x*exp( Sum_{k=2..n+1} A_{k}(x) ).
(4) exp(-A(x)) = G(x) where G(x*G(x)) = exp(-x) and G(-x) = e.g.f. of A087961.
a(n)=n!*T(n,1), T(n,m)=m/n*sum(T(n-m,k)*n^k/k!,k,1,n-m), n>m, T(n.n)=1. [Vladimir Kruchinin, May 06 2012]
Showing 1-3 of 3 results.

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