|
%I #3 Sep 02 2017 04:00:36
%S 1,3,33,519,9969,218907,5307201,139123215,3889995297,114928234611,
%T 3563543673825,115375173490839,3885328265571345,135675583665864843,
%U 4900856792035006593,182756242210436579871,7023982500750575903553,277842871320960134512611,11297961688442941015761825,471773677417286920645721895,20211597594930636918024401457,887652829316087359743197592315
%N G.f. A(x) satisfies: A(x - 4*x*A(x)) = x - x*A(x).
%F G.f. A(x) also satisfies:
%F (1) A(x) = (4/3)*Series_Reversion( x - 3*x*A(x) ) - x/3.
%F (2) A( (4*A(x) - x)/3 ) = (A(x) - x) / (4*A(x) - x).
%F a(n) = Sum_{k=0..n-1} A291820(n, k) * 4^k * 3^(n-k-1).
%e G.f.: A(x) = x + 3*x^2 + 33*x^3 + 519*x^4 + 9969*x^5 + 218907*x^6 + 5307201*x^7 + 139123215*x^8 + 3889995297*x^9 + 114928234611*x^10 +...
%e such that A(x - 4*x*A(x)) = x - x*A(x).
%e RELATED SERIES.
%e A(x - 4*x*A(x)) = x - x^2 - 3*x^3 - 33*x^4 - 519*x^5 - 9969*x^6 - 218907*x^7 - 5307201*x^8 +...
%e which equals x - x*A(x).
%e Series_Reversion( x - 4*x*A(x) ) = x + 4*x^2 + 44*x^3 + 692*x^4 + 13292*x^5 + 291876*x^6 + 7076268*x^7 + 185497620*x^8 +...
%e which equals (4/3)*A(x) - x/3.
%e A( (4*A(x) - x)/3 ) = x + 7*x^2 + 101*x^3 + 1919*x^4 + 42713*x^5 + 1058967*x^6 + 28469325*x^7 + 816617535*x^8 + 24729787889*x^9 + 784895219495*x^10 +...
%e which equals (A(x) - x) / (4*A(x) - x).
%o (PARI) {a(n) = my(A=x); for(i=1, n, A = (3/4)*serreverse( x - 4*x*A +x*O(x^n) ) + x/4 ); polcoeff(A, n)}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A291820, A291813, A291814, A291815, A291816, A291817, A291819.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Sep 02 2017
| 