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 1-2 of 2 results.

A375456 Expansion of g.f. A(x) satisfying x = A( A(x) - 2*A(x)^2*A'(x) ).

Original entry on oeis.org

1, 1, 5, 40, 414, 5096, 71465, 1113432, 18964415, 349252420, 6899717360, 145360352592, 3250782038728, 76887080836140, 1917401350590001, 50284361717695424, 1383636099826635216, 39865319955874291412, 1200467734347938040895, 37718141663144558046536, 1234556743772762830508484
Offset: 1

Views

Author

Paul D. Hanna, Sep 06 2024

Keywords

Comments

It appears that a(A219608(n)) is odd for n >= 1, and that the only other odd term is a(2) = 1.

Examples

			G.f.: A(x) = x + x^2 + 5*x^3 + 40*x^4 + 414*x^5 + 5096*x^6 + 71465*x^7 + 1113432*x^8 + 18964415*x^9 + 349252420*x^10 + ...
where x = A( A(x) - 2*A(x)^2 * A'(x) ).
RELATED SERIES.
Let R(x) be the series reversion of A(x) so that R(A(x)) = x, then
R(x) = x - x^2 - 3*x^3 - 20*x^4 - 190*x^5 - 2240*x^6 - 30759*x^7 - 475116*x^8 - 8081145*x^9 - 149243380*x^10 + ...
where R(x) = A(x) - 2*A(x)^2 * A'(x).
A(x)^2 = x^2 + 2*x^3 + 11*x^4 + 90*x^5 + 933*x^6 + 11420*x^7 + 158862*x^8 + 2453874*x^9 + ...

Links

Crossrefs

Cf. A219608, A259606.

Programs

  • PARI
    {a(n) = my(A=[0,1],Ax=x); for(i=1,n, A=concat(A,0); Ax=Ser(A);
A[#A] = -polcoeff( subst(Ax,x, Ax - 2*Ax^2*Ax')/2, #A-1); ); H=Ax; A[n+1]}
for(n=1,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) x = A( A(x) - 2*A(x)^2 * A'(x) ).
(2) A(A(x)) = x + 2*A(A(x))^2 * A'(A(x)).
(3) R(R(x)) = x - 2*x^2 * A'(R(x)), where A(R(x)) = x.
a(n) ~ c * d^n * n! * n^alpha, where d = 1.3534821142256737694364485294..., alpha = 2.7625501039589..., c = 0.0101323266748276... - Vaclav Kotesovec, Sep 07 2024

A360578 Expansion of g.f. A(x) satisfying A(x) = Series_Reversion( x - x*A'(x)*A(x) ).

Original entry on oeis.org

1, 1, 5, 42, 471, 6422, 101439, 1803949, 35459549, 760744865, 17651187689, 439893743313, 11711735210140, 331666197753372, 9954249177284539, 315638779480717743, 10545365878475964736, 370309787453143694246, 13637805276205022293179, 525684316153586923528166
Offset: 1

Views

Author

Paul D. Hanna, Feb 21 2023

Keywords

Examples

			G.f.: A(x) = x + x^2 + 5*x^3 + 42*x^4 + 471*x^5 + 6422*x^6 + 101439*x^7 + 1803949*x^8 + 35459549*x^9 + 760744865*x^10 + ...
such that A( x - x*A'(x)*A(x) ) = x.
Related series.
Series_Reversion(A(x)) = x - x^2 - 3*x^3 - 22*x^4 - 235*x^5 - 3153*x^6 - 49721*x^7 - 888784*x^8 - 17615520*x^9 + ...
A'(x)*A(x) = x + 3*x^2 + 22*x^3 + 235*x^4 + 3153*x^5 + 49721*x^6 + 888784*x^7 + 17615520*x^8 + ...
A(A(x)) = x + 2*x^2 + 12*x^3 + 110*x^4 + 1294*x^5 + 18127*x^6 + 290620*x^7 + 5206800*x^8 + 102633591*x^9 + ...
A'(A(x)) = 1 + 2*x + 17*x^2 + 208*x^3 + 3108*x^4 + 53328*x^5 + 1018948*x^6 + 21297818*x^7 + 481458997*x^8 + ...
A'(A(x))*A(A(x)) = x + 4*x^2 + 33*x^3 + 376*x^4 + 5242*x^5 + 84625*x^6 + 1534652*x^7 + 30682881*x^8 + ...

Links

Crossrefs

Cf. A259606, A213591, A303063.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A=serreverse(x - x*A'*A +x*O(x^n))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=x); for(i=1, n, A = x + sum(m=1, n, Dx(m-1, x^m*(A')^m*A^m/m!)) +O(x^(n+1))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); for(i=1, n, A = x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*(A')^m*A^m/m!)) +O(x^(n+1)))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
(1) A( x - x*A'(x)*A(x) ) = x.
(2) A(x) = x + A(x) * A'(A(x)) * A(A(x)).
(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n * A'(x)^n * A(x)^n / n!.
(4) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1) * A'(x)^n * A(x)^n / n! ).
a(n) ~ c * n! * n^alfa / LambertW(1)^n, where alfa = 1.5447806483693... and c = 0.02888888614196289496..., conjecture: alfa = 2*(2*LambertW(1) - 1 + 1/(1 + LambertW(1))). - Vaclav Kotesovec, Feb 22 2023
Showing 1-2 of 2 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.