|
%I #9 Sep 12 2017 23:29:54
%S 1,2,5,22,104,508,2581,13590,73255,402096,2240803,12645756,72120577,
%T 415017628,2406756231,14051487550,82524268241,487209739242,
%U 2889872479365,17213187356000,102916691619082,617446202168776,3715938753908449,22427371853933596,135714772335246022,823237636321437554,5004875492077075585,30490222425814170940
%N G.f. A(x) satisfies: A( x*A(x) - 2*x*A(x)^2 + x*A(x)^3 ) = x^2.
%H Paul D. Hanna, <a href="/A272821/b272821.txt">Table of n, a(n) for n = 1..300</a>
%e G.f.: A(x) = x + 2*x^2 + 5*x^3 + 22*x^4 + 104*x^5 + 508*x^6 + 2581*x^7 + 13590*x^8 + 73255*x^9 + 402096*x^10 + 2240803*x^11 + 12645756*x^12 +...
%e where A( x*A(x)*(1 - A(x))^2 ) = x^2.
%e RELATED SERIES.
%e A(x)^2 = x^2 + 4*x^3 + 14*x^4 + 64*x^5 + 321*x^6 + 1652*x^7 + 8718*x^8 + 47160*x^9 + 259848*x^10 + 1452340*x^11 + 8215412*x^12 +...
%e A(x)^3 = x^3 + 6*x^4 + 27*x^5 + 134*x^6 + 711*x^7 + 3846*x^8 + 21104*x^9 + 117600*x^10 + 663747*x^11 + 3785068*x^12 + 21775758*x^13 +...
%e A(x)^4 = x^4 + 8*x^5 + 44*x^6 + 240*x^7 + 1350*x^8 + 7664*x^9 + 43736*x^10 + 251408*x^11 + 1455577*x^12 + 8480432*x^13 + 49687828*x^14 +...
%e A(x) - 2*A(x)^2 + A(x)^4 = x - 2*x^3 + 3*x^5 - 12*x^7 + 39*x^9 - 130*x^11 + 495*x^13 - 1932*x^15 + 7818*x^17 +...
%e Let B(x) be the series reversion of g.f. A(x), so that A(B(x)) = x, then
%e B(x) = x - 2*x^2 + 3*x^3 - 12*x^4 + 39*x^5 - 130*x^6 + 495*x^7 - 1932*x^8 + 7818*x^9 - 32496*x^10 + 137158*x^11 - 587476*x^12 + 2544253*x^13 - 11117046*x^14 + 48960141*x^15 +...
%e where B(x^2) = x*A(x)*(1 - A(x))^2,
%e also, B(B(x)^2) = x*(1-x)^2 * B(x).
%e The square-root of x*A(x) is an integer series:
%e sqrt( x*A(x) ) = x + x^2 + 2*x^3 + 9*x^4 + 41*x^5 + 195*x^6 + 973*x^7 + 5063*x^8 + 27023*x^9 + 147147*x^10 + 814736*x^11 + 4573323*x^12 + 25964578*x^13 + 148836155*x^14 + 860244693*x^15 +...+ A292078(n)*x^n +...
%e Given g.f. A(x), the g.f. of A292078 equals G(x) such that
%e (1) G(x)^2 = A(x^2),
%e (2) G( x*G(x) - x*G(x)^3 ) = x^2.
%o (PARI) {a(n) = my(A=[1,2]); for(i=1,n, A=concat(A,0); F=x*Ser(A); A[#A] = -Vec( subst(F,x, x*F*(1-F)^2 ) )[#A] );A[n]}
%o for(n=1,40,print1(a(n),", "))
%Y Cf. A292078.
%K nonn
%O 1,2
%A _Paul D. Hanna_, May 09 2016
| 