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.

A374567 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2 + 2*x*A(x)^2 + 2*A(x)^3 ).

Original entry on oeis.org

1, 2, 9, 51, 325, 2222, 15926, 118085, 898217, 6970053, 54960439, 439112322, 3547096393, 28921270773, 237704587991, 1967321998468, 16381661824340, 137144132047520, 1153655788549216, 9746264972136632, 82656795697147384, 703459159019830315, 6005956718852682504, 51426768620398474939
Offset: 1

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Author

Paul D. Hanna, Aug 13 2024

Keywords

Comments

Compare to: C(x)^2 = C( x^2 - 2*C(x)^3 ), where C(x) = x - C(x)^2.

Examples

			G.f.: A(x) = x + 2*x^2 + 9*x^3 + 51*x^4 + 325*x^5 + 2222*x^6 + 15926*x^7 + 118085*x^8 + 898217*x^9 + 6970053*x^10 + ...
where A(x)^2 = A( x^2 + 2*x*A(x)^2 + 2*A(x)^3 ).
RELATED SERIES.
Let G(x) be the g.f. of the Wedderburn-Etherington numbers, then
A( x - x^2 - x*G(x) ) = x, where G(x) = x + (1/2)*(G(x)^2 + G(x^2)) begins
G(x) = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 11*x^7 + 23*x^8 + 46*x^9 + 98*x^10 + 207*x^11 + 451*x^12 + 983*x^13 + ... + A001190(n)*x^n + ...
A(x)^2 = x^2 + 4*x^3 + 22*x^4 + 138*x^5 + 935*x^6 + 6662*x^7 + 49191*x^8 + 373020*x^9 + 2887711*x^10 + 22727256*x^11 + ...
A(x)^3 = x^3 + 6*x^4 + 39*x^5 + 269*x^6 + 1938*x^7 + 14418*x^8 + 109932*x^9 + 854568*x^10 + 6747672*x^11 + ...
SPECIFIC VALUES.
A(t) = 1/5 at t = 0.1094430388151747748055350980742058560407673560783455...
where 1/25 = A( t^2 + 2*t/25 + 2/125 ).
A(t) = 1/6 at t = 0.1053569291935061227625330002451062383852684202941979...
where 1/36 = A( t^2 + t/18 + 1/108 ).
A(1/10) = 0.1471263013840628871589336795118257882025452972700045...
where A(1/10)^2 = A( 1/10^2 + (2/10)*A(1/10)^2 + 2*A(1/10)^3 ).
A(1/11) = 0.1237258078822115859596611191115342221543387518134407...
A(1/12) = 0.1081759735424269717469930892718654709953905803313352...

Links

Crossrefs

Cf. A374566, A271959, A001190.

Programs

  • PARI
    {a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A = concat(A, 0); Ax=Ser(A);
A[#A] = (1/2)*polcoeff( subst(Ax, x, x^2 + 2*x*Ax^2 + 2*Ax^3) - Ax^2, #A) ); 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) A(x)^2 = A( x^2 + 2*x*A(x)^2 + 2*A(x)^3 ).
(2) x = A( x - x^2 - x*G(x) ), where G(x) = x + (1/2)*(G(x)^2 + G(x^2)) is the g.f. of A001190, the Wedderburn-Etherington numbers.
(3) x^2 = A( x^2*(1 - G(x))^2 + 2*x^3 - x^4 ), where G(x) is the g.f. of A001190.
(4) x = A( x*sqrt(1 - 2*x - G(x^2)) - x^2 ), where G(x) is the g.f. of A001190.

A378247 G.f. A(x) satisfies A(x)^2 = A( x^2 + 2*x*A(x)^2 + 2*A(x)^4 ).

Original entry on oeis.org

1, 1, 3, 10, 39, 161, 699, 3135, 14427, 67716, 322959, 1560585, 7624007, 37593476, 186856061, 935214523, 4709265692, 23841104525, 121275951719, 619558165489, 3177346503440, 16351749778167, 84419824808865, 437105510426235, 2269266695980449, 11810014285000263, 61602685079710638
Offset: 1

Views

Author

Paul D. Hanna, Nov 20 2024

Keywords

Comments

Compare to C(x)^2 = C( x^2 + 2*x*C(x)^2 ) where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

Examples

			G.f.: A(x) = x + x^2 + 3*x^3 + 10*x^4 + 39*x^5 + 161*x^6 + 699*x^7 + 3135*x^8 + 14427*x^9 + 67716*x^10 + 322959*x^11 + 1560585*x^12 + ...
where A(x)^2 = A( x^2 + 2*x*A(x)^2 + 2*A(x)^4 ).
RELATED SERIES.
Let G(x) be the g.f. of the Wedderburn-Etherington numbers, then
A( x - x^2 - x*G(x^2) ) = x, where G(x) = x + (1/2)*(G(x)^2 + G(x^2)) begins
G(x) = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 11*x^7 + 23*x^8 + 46*x^9 + 98*x^10 + 207*x^11 + 451*x^12 + 983*x^13 + 2179*x^14 + ... + A001190(n)*x^n + ...
Let B(x) be the series reversion of g.f. A(x) so that B(A(x)) = x, then
B(x) = x - x^2 - x^3 - x^5 - x^7 - 2*x^9 - 3*x^11 - 6*x^13 - 11*x^15 - 23*x^17 - 46*x^19 - 98*x^21 - 207*x^23 + ...
where B(x) = x - x^2 - x*G(x^2).
SPECIFIC VALUES.
A(t) = 1/3 at t = 0.1804894059505127351310871071614416167035910065610113327...
  where 1/9 = A( t^2 + 2*t/9 + 2/81 ).
A(t) = 1/4 at t = 0.1708289565101545485579649480920097855916395263217351536...
  where 1/16 = A( t^2 + t/8 + 1/128 ).
A(t) = 1/5 at t = 0.1516661092515691718015998101146470241027491658579501286...
  where 1/25 = A( t^2 + 2*t/25 + 2/625 ).
A(t) = 1/6 at t = 0.1341268789797555579297424694390747929782019601987848246...
  where 1/36 = A( t^2 + t/18 + 1/648 ).
A(1/6) = 0.2368314953172156547771056118501694080205525703518284958...
A(1/7) = 0.1824082884402163049324182135107985537409785918465705698...
A(1/8) = 0.1515179821748020682616541846638756124979071552818869937...
A(1/9) = 0.1303577455916869424988611259176631850931169441135101392...
A(1/10) = 0.1146797533131163787803333792504789207692884367435306666...

Links

Crossrefs

Cf. A001190, A374567, A271959, A271960, A374566.

Programs

  • PARI
    {a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A = concat(A, 0); Ax=Ser(A);
A[#A] = (1/2)*polcoeff( subst(Ax, x, x^2 + 2*x*Ax^2 + 2*Ax^4) - Ax^2, #A) ); 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 in which G(x) = x + (1/2)*(G(x)^2 + G(x^2)) is the g.f. of A001190, the Wedderburn-Etherington numbers.
(1) A(x)^2 = A( x^2 + 2*x*A(x)^2 + 2*A(x)^4 ).
(2) x = A( x - x^2 - x*G(x^2) ).
(3) x = A( x + x^2 - 2*x*G(x) + x*G(x)^2 ).
(4) x = A( x*sqrt(1 - 2*x^2 - G(x^4)) - x^2 ).
(5) x^2 = A( x^2*((1 - G(x))^2 + 2*x)^2 + x^4 ).
(6) G(A(x)) = 1 - sqrt(x/A(x) - A(x)).
Showing 1-2 of 2 results.

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