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URL: https://oeis.org/A392207

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A392207
G.f. satisfies A(x) = A( x^3 + 2*x*A(x)^3 ) / A( x^2 + x*A(x)^2 ).
1
1, 1, 2, 7, 27, 119, 548, 2637, 13026, 65780, 337707, 1757578, 9250051, 49145958, 263237032, 1419895905, 7706133564, 42050981511, 230576847319, 1269803582041, 7020248329269, 38949655525029, 216796208810075, 1210253107814276, 6774412445878257
OFFSET
1,3
COMMENTS
Compare g.f. to C(x) = C( x^3 + 3*x*C(x)^3 ) / C( x^2 + 2*x*C(x)^2 ), where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
LINKS
FORMULA
a(n) ~ c * d^n / n^(3/2), where d = 5.9574901942727101173439965971423618354705087738... and c = 0.0346212142580246250658945905427864262460458... - Vaclav Kotesovec, Jan 27 2026
EXAMPLE
G.f.: A(x) = x + x^2 + 2*x^3 + 7*x^4 + 27*x^5 + 119*x^6 + 548*x^7 + 2637*x^8 + 13026*x^9 + 65780*x^10 + 337707*x^11 + 1757578*x^12 + ...
RELATED SERIES.
A( x^2 + x*A(x)^2 ) = x^2 + x^3 + 3*x^4 + 7*x^5 + 25*x^6 + 92*x^7 + 395*x^8 + 1775*x^9 + 8434*x^10 + 41296*x^11 + 207548*x^12 + ...
A( x^3 + 2*x*A(x)^3 ) = x^3 + 2*x^4 + 6*x^5 + 19*x^6 + 72*x^7 + 298*x^8 + 1334*x^9 + 6260*x^10 + 30458*x^11 + 152057*x^12 + ...
where A(x) = A( x^3 + 2*x*A(x)^3 ) / A( x^2 + x*A(x)^2 ).
SPECIFIC VALUES.
A(1/6) = 0.2340131020603190557774441656651291677871842374865636...
A(1/7) = 0.1770301578018354684471813891979108108611537179572220...
A(1/8) = 0.1482113503746846549016319574844383133115150395611583...
A(1/9) = 0.1282026744951353772228404626466647470632908369974957...
A(1/10) = 0.113196901049813303366935566357283343157862454102091...
A(-1/6) = -0.144760575655613622506042473975422527307143856859108...
A(t) = 1/5 at t = 0.154808464112305890755432003988431981924537782172...
A(t) = 1/6 at t = 0.136758859393932287411463646383268564071478898268...
A(t) = 1/7 at t = 0.121393199796082954259527612707777740647224296563...
A(t) = 1/8 at t = 0.108787611078989381888009921008915095677710375617...
PROG
(PARI) {a(n) = my(A = x); for(k=1, n, A = truncate(A) +x^3*O(x^k);
A = subst(A, x, x^3 + 2*x*A^3) / subst(A, x, x^2 + x*A^2) ); polcoef(A, n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
Sequence in context: A036783 A368762 A150645 * A060017 A211475 A213226
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 12 2026
STATUS
approved