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.

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A171212 G.f.: A(x) satisfies A(x) = x + x*A(A(2*x)).

Original entry on oeis.org

0, 1, 2, 16, 320, 12928, 985088, 140861440, 38451150848, 20403322617856, 21307854867660800, 44110759073910095872, 181739941085108158595072, 1493546441998961207249207296, 24512116566896662943648857456640
Offset: 0

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Author

Paul D. Hanna, Dec 08 2009

Keywords

Comments

More generally, if F(x) = x + x*F(F(qx)), then
F(x) = x + x*F(qx) + x*F(qx)*F(qF(qx)) + x*F(qx)*F(qF(qx))*F(qF(qF(qx))) +...
with a simple solution at q=1/2:
F(x) = x/(1-x/2) satisfies: F(x) = x + x*F(F(x/2)).

Examples

			G.f.: A(x) = x + 2*x^2 + 16*x^3 + 320*x^4 + 12928*x^5 +...
A(A(x)) = x + 4*x^2 + 40*x^3 + 808*x^4 + 30784*x^5 + 2200960*x^6 +...+ a(n)*x^n/2^(n-1) +...
As a formal series involving products of iterations of the g.f.,
A(x) = x + x*A(2x) + x*A(2x)*A(2A(2x)) + x*A(2x)*A(2A(2x))*A(2A(2A(2x))) +...
which, upon replacing x with A(2x), yields:
A(A(2x)) = A(2x) + A(2x)*A(2A(2x)) + A(2x)*A(2A(2x))*A(2A(2A(2x))) +...
thus A(x) = x + x*A(A(2x)).

Crossrefs

Cf. A171213 (q=3), A171214 (q=1/3).

Programs

  • PARI
    {a(n,q=2)=local(A=x+x^2);for(i=1,n,A=x+x*subst(A,x,subst(A,x,q*x+O(x^n))));polcoeff(A,n)}

Extensions

a(0) = 0 added by Jason Yuen, Feb 07 2025

A171213 G.f.: A(x) satisfies A(x) = x + x*A(A(3*x)).

Original entry on oeis.org

1, 3, 54, 3402, 618921, 318392208, 474852630879, 2094575471899362, 27570620677894020891, 1086589159409074932937593, 128377839432663886566934695744, 45490432595875817814676362194769627
Offset: 1

Views

Author

Paul D. Hanna, Dec 08 2009

Keywords

Comments

More generally, if F(x) = x + x*F(F(qx)), then
F(x) = x + x*F(qx) + x*F(qx)*F(qF(qx)) + x*F(qx)*F(qF(qx))*F(qF(qF(qx))) +...
with a simple solution at q=1/2:
F(x) = x/(1-x/2) satisfies: F(x) = x + x*F(F(x/2)).

Examples

			G.f.: A(x) = x + 3*x^2 + 54*x^3 + 3402*x^4 + 618921*x^5 +...
A(A(x)) = x + 6*x^2 + 126*x^3 + 7641*x^4 + 1310256*x^5 +...+ a(n)*x^n/3^(n-1) +...
As a formal series involving products of iterations of the g.f.,
A(x) = x + x*A(3x) + x*A(3x)*A(3A(3x)) + x*A(3x)*A(3A(3x))*A(3A(3A(3x))) +...
which, upon replacing x with A(3x), yields:
A(A(3x)) = A(3x) + A(3x)*A(3A(3x)) + A(3x)*A(3A(3x))*A(3A(3A(3x))) +...
thus A(x) = x + x*A(A(3x)).

Crossrefs

Cf. A171212 (q=2), A171214 (q=1/3).

Programs

  • PARI
    {a(n,q=3)=local(A=x+x^2);for(i=1,n,A=x+x*subst(A,x,subst(A,x,q*x+O(x^n))));polcoeff(A,n)}
Showing 1-2 of 2 results.

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

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