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A378255
G.f. A(x) satisfies A(x) = x + A(A(x)^2) + A(A(x)^3).
1
1, 1, 3, 11, 44, 193, 882, 4178, 20305, 100694, 507493, 2591897, 13384911, 69773480, 366661360, 1940336952, 10331179153, 55306072496, 297499878304, 1607212566176, 8716586959731, 47440220540922, 259021671704538, 1418386679682870, 7787843448380598, 42865830110488341, 236480195092162079
OFFSET
1,3
LINKS
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x) = x + A(A(x)^2) + A(A(x)^3).
(2) A(x) = Series_Reversion(x - A(x^2) - A(x^3)).
(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (A(x^2) + A(x^3))^n / n!.
(4) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (A(x^2) + A(x^3))^n/x / n! ).
EXAMPLE
G.f.: A(x) = x + x^2 + 3*x^3 + 11*x^4 + 44*x^5 + 193*x^6 + 882*x^7 + 4178*x^8 + 20305*x^9 + 100694*x^10 + 507493*x^11 + 2591897*x^12 + ...
where A(x) = x + A(A(x)^2) + A(A(x)^3).
RELATED SERIES.
A(A(x)^2) = x^2 + 2*x^3 + 8*x^4 + 32*x^5 + 140*x^6 + 642*x^7 + 3044*x^8 + 14810*x^9 + 73508*x^10 + 370744*x^11 + 1894641*x^12 + ...
A(A(x)^3) = x^3 + 3*x^4 + 12*x^5 + 53*x^6 + 240*x^7 + 1134*x^8 + 5495*x^9 + 27186*x^10 + 136749*x^11 + 697256*x^12 + ...
A(x^2) + A(x^3) = x^2 + x^3 + x^4 + 4*x^6 + 11*x^8 + 3*x^9 + 44*x^10 + 204*x^12 + 882*x^14 + 44*x^15 + 4178*x^16 + 20498*x^18 + 100694*x^20 + 882*x^21 + 507493*x^22 + 2596075*x^24 + ...
where A(x - A(x^2) - A(x^3)) = x.
SPECIFIC VALUES.
A(t) = 2/7 at t = 0.17113633000646334369180481612349578624783801172280...
where t = 2/7 - A(4/49) - A(8/343).
A(t) = 1/4 at t = 0.16675404370583413662402450218565513714831350182372...
where t = 1/4 - A(1/16) - A(1/64).
A(t) = 1/5 at t = 0.15010878278748588281213370320635881316994691322045...
where t = 1/5 - A(1/25) - A(1/125).
A(t) = 1/6 at t = 0.13339423917640795169936128266096825782919420931151...
where t = 1/6 - A(1/36) - A(1/216).
A(t) = 1/7 at t = 0.11908088070152938397080640031639630333627904913531...
A(t) = 1/8 at t = 0.10716175341844903845196692277233592851173749413122...
A(t) = 1/9 at t = 0.09723347082793633309633745814458884367513391389109...
A(t) = 1/10 at t = 0.0888958823866966406913909794974554201549795441139...
A(1/16) = 0.0673646719229547504984252010615186334929817484538318...
A(1/25) = 0.0418256346622057039599677785843885345768765252605719...
A(1/36) = 0.0286210615541792671675075085528989453238496370093068...
A(1/64) = 0.0158812843712111128775502967528262293587047497224419...
A(1/125) = 0.0080655825503084132278985182092526522531765615189742...
A(1/216) = 0.0046513659360794477997978754527994635136228203458472...
PROG
(PARI) \\ Formula: A(x) = Series_Reversion(x - A(x^2) - A(x^3)) \\
{a(n) = my(A=x+x^2); for(i=0, n, A = serreverse(x - subst(A, x, x^2 +x^2*O(x^n)) - subst(A, x, x^3 +x^2*O(x^n)))) ; polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) \\ Formula: A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (A(x^2) + A(x^3))^n/n! \\
{Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=x+x^2+x*O(x^n)); for(i=1, n, A = x + sum(m=1, n, Dx(m-1, (subst(A, x, x^2+x*O(x^n)) + subst(A, x, x^3+x*O(x^n)))^m)/m!)+x*O(x^n)); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
Cf. A141200.
Sequence in context: A256752 A018963 A217886 * A234627 A030815 A030880
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 23 2024
STATUS
approved