login
G.f. satisfies: 7*A(x) = 6 + x + A(x)^6, starting with [1,1,15].
2

%I #7 Nov 28 2017 03:51:46

%S 1,1,15,470,18390,805806,37828981,1860433080,94614523740,

%T 4935081398830,262560448214031,14193030016877406,777315341935068820,

%U 43039297954660894560,2405249540028525971070,135492504636185052358656

%N G.f. satisfies: 7*A(x) = 6 + x + A(x)^6, starting with [1,1,15].

%C See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.

%F G.f.: A(x) = 1 + Series_Reversion(1+7*x - (1+x)^6). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(6*n,n)/(5*n+1) * (6+x)^(5*n+1)/7^(6*n+1). - _Paul D. Hanna_, Jan 24 2008

%F a(n) ~ (-6 + 5*(7/6)^(6/5))^(1/2 - n) / (2^(3/5) * 3^(1/10) * 7^(2/5) * n^(3/2) * sqrt(5*Pi)). - _Vaclav Kotesovec_, Nov 28 2017

%e A(x) = 1 + x + 15*x^2 + 470*x^3 + 18390*x^4 + 805806*x^5 +...

%e A(x)^6 = 1 + 6*x + 105*x^2 + 3290*x^3 + 128730*x^4 + 5640642*x^5 +...

%t CoefficientList[1 + InverseSeries[Series[1+7*x - (1+x)^6, {x, 0, 20}], x], x] (* _Vaclav Kotesovec_, Nov 28 2017 *)

%o (PARI) {a(n)=local(A=1+x+15*x^2+x*O(x^n));for(i=0,n,A=A-7*A+6+x+A^6);polcoeff(A,n)}

%Y Cf. A120588 - A120599, A120601 - A120607.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jun 16 2006