A272460 G.f. A(x) satisfies: A( x - A(x^3)/x ) = x.
1, 1, 2, 5, 15, 49, 168, 596, 2170, 8063, 30451, 116545, 451038, 1762065, 6939684, 27523374, 109832228, 440668881, 1776599145, 7193526536, 29240389629, 119276102017, 488106369196, 2003299984825, 8244088853598, 34010402405020, 140627814353509, 582704045483909, 2419228983607503, 10062353339406026, 41924039720446064, 174952464642171681, 731184941189099208, 3060168941260579386
Offset: 1
Keywords
Examples
G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 15*x^5 + 49*x^6 + 168*x^7 + 596*x^8 + 2170*x^9 + 8063*x^10 + 30451*x^11 + 116545*x^12 +... where A( x - A(x^3)/x ) = x. RELATED SERIES. Let B(x) be the series reversion of g.f. A(x), A(B(x)) = x, then B(x) = x - x^2 - x^5 - 2*x^8 - 5*x^11 - 15*x^14 - 49*x^17 - 168*x^20 - 596*x^23 - 2170*x^26 - 8063*x^29 - 30451*x^32 - 116545*x^35 +... such that B(x) = x - A(x^3)/x.
Links
- Paul D. Hanna, Table of n, a(n) for n = 1..300
Programs
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PARI
{a(n) = my(A=x); for(i=1,n, A = serreverse( x - subst(A,x,x^3 +x^3*O(x^n))/x )); polcoeff(A,n)} for(n=1,50,print1(a(n),", "))
Formula
a(n) ~ c * d^n / n^(3/2), where d = 4.3788729685558146277374586... and c = 0.0933818743555997288781743... . - Vaclav Kotesovec, May 03 2016