login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

G.f. A(x) satisfies: A( A(x)^2 - 2*A(x)^3 ) = x*A(x).
1

%I #7 May 13 2016 19:44:02

%S 1,2,6,28,150,848,4988,30320,189030,1201792,7761296,50774280,

%T 335776204,2241022944,15075584864,102113820256,695842808070,

%U 4767068036992,32813466593552,226829615262744,1574027068685944,10960566883613504,76564394477117688,536383955787886208,3767701071640753708,26530029714589074464,187231524430927518240

%N G.f. A(x) satisfies: A( A(x)^2 - 2*A(x)^3 ) = x*A(x).

%H Paul D. Hanna, <a href="/A272820/b272820.txt">Table of n, a(n) for n = 1..300</a>

%F G.f. A(x) satisfies: A( A(x^2 - 2*x^3) / x ) = x.

%e G.f.: A(x) = x + 2*x^2 + 6*x^3 + 28*x^4 + 150*x^5 + 848*x^6 + 4988*x^7 + 30320*x^8 + 189030*x^9 + 1201792*x^10 +...

%e where A( A(x)^2 - 2*A(x)^3 ) = x*A(x).

%e RELATED SERIES.

%e A(x)^2 = x^2 + 4*x^3 + 16*x^4 + 80*x^5 + 448*x^6 + 2632*x^7 + 15952*x^8 + 99168*x^9 + 629184*x^10 + 4057272*x^11 + 26511544*x^12 +...

%e A(x)^3 = x^3 + 6*x^4 + 30*x^5 + 164*x^6 + 966*x^7 + 5904*x^8 + 36924*x^9 + 235248*x^10 + 1522086*x^11 + 9974080*x^12 + 66055800*x^13 +...

%e A(x)^2 - 2*A(x)^3 = x^2 + 2*x^3 + 4*x^4 + 20*x^5 + 120*x^6 + 700*x^7 + 4144*x^8 + 25320*x^9 + 158688*x^10 + 1013100*x^11 + 6563384*x^12 +...

%e A(x^2 - 2*x^3) = x^2 - 2*x^3 + 2*x^4 - 8*x^5 + 14*x^6 - 36*x^7 + 100*x^8 - 272*x^9 + 822*x^10 - 2396*x^11 + 7296*x^12 - 22176*x^13 + 67868*x^14 +...

%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 + 2*x^3 - 8*x^4 + 14*x^5 - 36*x^6 + 100*x^7 - 272*x^8 + 822*x^9 - 2396*x^10 + 7296*x^11 - 22176*x^12 + 67868*x^13 +...

%e where B(x) = A(x^2 - 2*x^3)/x,

%e also, B( x*B(x) ) = x^2 - 2*x^3.

%o (PARI) {a(n) = my(A=[1],F); for(i=1,n, A = concat(A,0); F = x*Ser(A); A[#A] = -Vec(subst(F,x, F^2 - 2*F^3))[#A]); A[n]}

%o for(n=1,30,print1(a(n),", "))

%K nonn

%O 1,2

%A _Paul D. Hanna_, May 12 2016