A291818 G.f. A(x) satisfies: A(x - 4*x*A(x)) = x - x*A(x).

1, 3, 33, 519, 9969, 218907, 5307201, 139123215, 3889995297, 114928234611, 3563543673825, 115375173490839, 3885328265571345, 135675583665864843, 4900856792035006593, 182756242210436579871, 7023982500750575903553, 277842871320960134512611, 11297961688442941015761825, 471773677417286920645721895, 20211597594930636918024401457, 887652829316087359743197592315

Offset

1,2

Links

Table of n, a(n) for n=1..22.

Formula

G.f. A(x) also satisfies:

(1) A(x) = (4/3)*Series_Reversion( x - 3*x*A(x) ) - x/3.

(2) A( (4*A(x) - x)/3 ) = (A(x) - x) / (4*A(x) - x).

a(n) = Sum_{k=0..n-1} A291820(n, k) * 4^k * 3^(n-k-1).

Example

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 +...
such that  A(x - 4*x*A(x)) = x - x*A(x).
RELATED SERIES.
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 +...
which equals x - x*A(x).
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 +...
which equals (4/3)*A(x) - x/3.
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 +...
which equals (A(x) - x) / (4*A(x) - x).

Prog

(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)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A291820, A291813, A291814, A291815, A291816, A291817, A291819.

Sequence in context: A336539 A221147 A355795 * A209245 A092170 A083080

Adjacent sequences: A291815 A291816 A291817 * A291819 A291820 A291821

Keyword

nonn

Author

Paul D. Hanna, Sep 02 2017

Status

approved