A372526 Expansion of g.f. A(x) satisfying A( x*A(x)^2 + A(x)^4 ) = A(x)^3.
1, 1, 2, 6, 20, 70, 256, 969, 3762, 14895, 59916, 244179, 1006026, 4183396, 17534888, 74007851, 314256048, 1341575769, 5754629794, 24789907450, 107202369386, 465209278326, 2025212712660, 8842042378050, 38707067608872, 169860383434800, 747096961093560, 3292855742992644
Offset: 1
Keywords
Examples
G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 70*x^6 + 256*x^7 + 969*x^8 + 3762*x^9 + 14895*x^10 + 59916*x^11 + 244179*x^12 + ... where A( x*A(x)^2 + A(x)^4 ) = A(x)^3. RELATED SERIES. (1) A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 31*x^6 + 114*x^7 + 432*x^8 + 1676*x^9 + 6633*x^10 + 26676*x^11 + 108696*x^12 + ... (2) x*A(x)^2 + A(x)^4 = x^3 + 3*x^4 + 9*x^5 + 30*x^6 + 108*x^7 + 405*x^8 + 1560*x^9 + 6138*x^10 + 24570*x^11 + 99738*x^12 + ... (3) Let R(x) be the series reversion of A(x), R(A(x)) = x, then R(x) = x - x^2 - x^4 - x^10 - x^28 - x^82 - x^244 - x^730 + ... + -x^(3^n+1) + ... SPECIFIC VALUES. A(1/5) = 0.2937167157779136500722875625899113632023... A(1/6) = 0.2150539986528250703029216090552606059919... A(1/7) = 0.1740789503092637057579787813575613522976... A(1/8) = 0.1471095742959948638409574049543396207684...
Links
- Paul D. Hanna, Table of n, a(n) for n = 1..1030
Programs
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PARI
/* Using series reversion of x - x*Sum_{n>=0} x^(3^n) */ {a(n) = my(A); A = serreverse( x - x*sum(k=0, ceil(log(n)/log(3)), x^(3^k) +x*O(x^n)) ); polcoeff(A, n)} for(n=1, 35, print1(a(n), ", ")) -
PARI
/* Using A(x)^3 = A( x*A(x)^2 + A(x)^4 ) */ {a(n) = my(A=[1], F); for(i=1, n, A = concat(A, 0); F = x*Ser(A); A[#A] = polcoeff( subst(F, x, x*F^2 + F^4 ) - F^3, #A+2) ); A[n]} for(n=1, 35, print1(a(n), ", "))
Formula
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^3 = A( x*A(x)^2 + A(x)^4 ).
(2) A(x)^9 = A( x*A(x)^8 + A(x)^10 + A(x)^12 ).
(3) A(x)^27 = A( x*A(x)^26 + A(x)^28 + A(x)^30 + A(x)^36 ).
(4) A(x)^(3^n) = A( x*A(x)^(3^n-1) + Sum_{k=0..n-1} A(x)^(3^n+3^k) ) for n > 0.
(5) A(x) = x + Sum_{n>=0} A(x)^(3^n+1).
(6) A(x) = Series_Reversion(x - x*Sum_{n>=0} x^(3^n) ).
The radius of convergence r and A(r) satisfy: 1 = Sum_{n>=0} (3^n+1) * A(r)^(3^n) and r = A(r) - Sum_{n>=0} A(r)^(3^n+1), where r = 0.214732801800375010254079407876131682823903064701286670006... and A(r) = 0.384967312289976324530970877165834568783164468488676531438...