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.

A075864 G.f. satisfies A(x) = 1 + Sum_{n>=0} (x*A(x))^(2^n).

Original entry on oeis.org

1, 1, 2, 4, 10, 26, 72, 204, 594, 1762, 5318, 16270, 50360, 157392, 496016, 1574432, 5028962, 16152194, 52133154, 169004450, 550036778, 1796512970, 5886709502, 19346204982, 63751851400, 210605429496, 697337388556, 2313871053172, 7692939444640, 25623793107344
Offset: 0

Views

Author

Paul D. Hanna, Oct 15 2002

Keywords

Comments

Number of Dyck n-paths with all ascent lengths being a power of 2. - David Scambler, May 07 2012

Examples

			G.f. (offset 0): A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 26*x^5 + 72*x^6 + 204*x^7 + 594*x^8 + 1762*x^9 + 5318*x^10 + ...
SPECIFIC VALUES.
The following values are for the g.f. at offset 1: A(x) = x + x^2 + 2*x^3 + 4*x^4 + 10*x^5 + 26*x^6 + 72*x^7 + 204*x^8 + ...
A(t) = 1/2 at t = 0.275266504782383938866239471561026684712237255315...
where 1/4 = A( (1/2)*t/(1-t) ) and t = (1/2)/(1 + Sum_{n>=0} 1/2^(2^n)).
A(t) = 1/3 at t = 0.228789618697442059759075468255467011039543924763...
where 1/9 = A( (1/3)*t/(1-t) ) and t = (1/3)/(1 + Sum_{n>=0} 1/3^(2^n)).
A(1/4) = 0.392935121163880589695619242847181861583875787578...
where A(1/4)^2 = A( (1/3)*A(1/4) ).
A(1/5) = 0.269480257065638376643289111191173593741789085897...
where A(1/5)^2 = A( (1/4)*A(1/5) ).
A(1/6) = 0.209130395397987995845331540196686970439063098884...
where A(1/6)^2 = A( (1/5)*A(1/6) ).

Links

Crossrefs

Cf. A075853, A374565.

Programs

  • Maple
    b:= proc(x, y, t) option remember; `if`(x<0 or y>x, 0,
      `if`(x=0, 1, b(x-1, y+1, true)+`if`(t, add(
       b(x-2^j, y-2^j, false), j=0..ilog2(y)), 0)))
    end:
a:= n-> b(2*n, 0, true):
seq(a(n), n=0..32);  # Alois P. Heinz, Apr 01 2019
  • Mathematica
    seq = {};
f[x_, y_, d_] :=
  f[x, y, d] =
   If[x < 0 || y < x , 0,
    If[x == 0 && y == 0, 1,
     f[x - 1, y, 0] + f[x, y - If[d == 0, 1, d], If[d == 0, 1, 2*d]]]];
For[n = 0, n <= 27, n++, seq = Append[seq, f[n, n, 0]]]; seq
(* David Scambler, May 07 2012 *)
A[_] = 0; m = 32;
Do[A[x_] = 1+Sum[(x A[x])^(2^n)+O[x]^m, {n, 0, Log[2, m]//Ceiling}], {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, May 20 2022 *)
  • PARI
    N=66; K=ceil(log(N)/log(2))+1; x='x+O('x^N); Vec(serreverse(x/(1 + sum(k=0,K,x^(2^k) ) ) ) ) \\ Joerg Arndt, Apr 01 2019
  • PARI
    {a(n) = my(A=[1], Ax);
for(i=1, n, A=concat(A, 0); Ax=x*Ser(A);
A[#A] = -polcoeff( Ax^2 - subst(Ax,x, x*Ax/(1-x) ), #A+1) ); A[n]}
for(n=1, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Jul 12 2024

Formula

G.f. A(x) satisfies x*A(x) = series_reversion( x / ( 1 + Sum_{k>=0} x^(2^k) ) ). - Joerg Arndt, Apr 01 2019
From Paul D. Hanna, Jul 12 2024: (Start)
G.f. A(x) = x*Sum_{n>=0} a(n)*x^n (offset 1) satisfies the following formulas.
(1) A(x)^2 = A( x*A(x)/(1-x) ).
(2) A(x)^4 = A( x*A(x)^3/(1 - x - x*A(x)) ).
(3) A(x)^8 = A( x*A(x)^7/(1 - x - x*A(x) - x*A(x)^3) ).
(4) A(x)^(2^n) = A( x*A(x)^(2^n-1) / (1 - x*Sum_{k=0..n-1} A(x)^(2^k-1)) ) for n >= 1.
The radius of convergence r and A(r) satisfy r = 1/(Sum_{n>=0} 2^n*A(r)^(2^n-1)) and A(r) = A( A(r)*r/(1-r) )^(1/2), where r = 0.285128929740568796881205193649402054331317007180873... and A(r) = 0.621954965556741102287309027445345554104820417676869...
(End)

A380554 G.f. A(x) satisfies A(x)^4 = A( A(x)^3 * x/(1-x) ).

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 16, 36, 75, 163, 391, 991, 2498, 6150, 15016, 37116, 93482, 238154, 608074, 1551370, 3964200, 10176384, 26261500, 68034484, 176661828, 459534596, 1197777556, 3129475636, 8195867902, 21508247446, 56540427826, 148863643466, 392539322259, 1036662269875, 2741706892035
Offset: 1

Views

Author

Paul D. Hanna, Jan 26 2025

Keywords

Examples

			G.f.: A(x) = x + x^2 + x^3 + x^4 + 2*x^5 + 6*x^6 + 16*x^7 + 36*x^8 + 75*x^9 + 163*x^10 + 391*x^11 + 991*x^12 + 2498*x^13 + 6150*x^14 + 15016*x^15 + ...
where A(x)^4 = A( A(x)^3 * x/(1-x) );
also, A(x) = x*(1 + A(x) + A(x)^4 + A(x)^16 + A(x)^64 + ...).
RELATED SERIES.
x/(1 + x + x^4 + x^16 + x^64 + ...) = x - x^2 + x^3 - x^4 + x^6 - 2*x^7 + 3*x^8 - 3*x^9 + 2*x^10 - 3*x^12 + 6*x^13 - 8*x^14 + 8*x^15 - 5*x^16 + ...
where x = A( x/(1 + x + x^4 + x^16 + x^64 + ...) ).
A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 10*x^6 + 18*x^7 + 42*x^8 + 112*x^9 + 288*x^10 + ...
A(x)^4 = x^4 + 4*x^5 + 10*x^6 + 20*x^7 + 39*x^8 + 88*x^9 + 228*x^10 + 600*x^11 + ...
SPECIFIC VALUES.
A(t) = 7/10 at t = 0.36018915820185609929548309671397017657231396...
  where (7/10)^4 = A( (7/10)^3*t/(1-t) )
  and t = (7/10)/(1 + Sum_{n>=0} (7/10)^(4^n)).
A(t) = 2/3 at t = 0.357324077294579321123715825007257976292387856...
  where 16/81 = A( (8/27)*t/(1-t) )
  and t = (2/3)/(1 + Sum_{n>=0} (2/3)^(4^n)).
A(t) = 1/2 at t = 0.319996875030517280093584464262123092506355813...
  where 1/16 = A( (1/8)*t/(1-t) )
  and t = (1/2)/(1 + Sum_{n>=0} (1/2)^(4^n)).
A(t) = 1/3 at t = 0.247706417742171319902767393551872977585317906...
  where 1/81 = A( (1/27)*t/(1-t) )
  and t = (1/3)/(1 + Sum_{n>=0} (1/3)^(4^n)).
A(1/3) = 0.54373202136840396341881074287828877295481851718413...
  where A(1/3)^4 = A( A(1/3)^3*(1/2) ).
A(1/4) = 0.33766677567921691723942758840979376280294197783058...
  where A(1/4)^4 = A( A(1/4)^3*(1/3) ).
A(1/5) = 0.25099215755350299738032710744403195608988446686839...
A(1/6) = 0.20032206620931060989695576481191496886558371212657...

Links

Crossrefs

Cf. A075864, A374565.

Programs

  • PARI
    /* A(x) = Series_Reversion( x/(1 + Sum_{n>=0} x^(4^n)) ) */
{a(n) = my(A = serreverse( x/(1 + sum(m=0,ceil(log(n+1)/log(4)), x^(4^m) +x*O(x^n))) )); polcoef(A,n)}
for(n=1, 40, print1(a(n), ", "))
  • PARI
    /* A(x)^4 = A( A(x)^3 * x/(1-x) ) */
{a(n) = my(A=[1], Ax);
for(i=1, n, A=concat(A, 0); Ax=x*Ser(A);
A[#A] = -polcoeff( Ax^4 - subst(Ax, x, Ax^3*x/(1-x) ), #A+3) ); A[n]}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x) = Series_Reversion( x/(1 + Sum_{n>=0} x^(4^n)) ).
(2) A(x) = x * (1 + Sum_{n>=0} A(x)^(4^n)).
(3) A(x) = x/(1-x) * (1 + Sum_{n>=1} A(x)^(4^n)).
(4) A(x)^4 = A( A(x)^3 * x/(1-x) ).
(5) A(x)^16 = A( A(x)^15 * x/(1 - x - x*A(x)^3) ).
(6) A(x)^64 = A( A(x)^63 * x/(1 - x - x*A(x)^3 - x*A(x)^15) ).
(7) A(x)^(4^n) = A( A(x)^(4^n-1) * x/(1 - x*Sum_{k=0..n-1} A(x)^(4^k-1)) ) for n >= 1.
The radius of convergence r and A(r) satisfy r = 1/(Sum_{n>=0} 4^n*A(r)^(4^n-1)) and A(r) = A( A(r)^3*r/(1-r) )^(1/4), where r = 0.3613437470225014946622689597447779556234350427479140... and A(r) = 0.7371720020640001613320630406857895231048184830453856...
Showing 1-2 of 2 results.

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