A195195 G.f. A(x) satisfies: A( x - x*A(2*x)/2 ) = x.

1, 1, 4, 31, 412, 9322, 370792, 26998435, 3721060012, 992975985526, 520211594819608, 539421251144944822, 1112261291753364074008, 4572473903615507238126100, 37530785839971207229105541584, 615542030204664293118535562855563, 20181157288253257322062384770648622156

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..50

Formula

G.f.: A(x) = INV(x - x*INV(x - 2*x*INV(x - 2^2*x*INV(x - 2^3*x*INV(x - 2^4*x*INV(x - ...)))))), where INV(F(x)) = series reversion of F(x).

G.f. A(x) also satisfies [from Paul D. Hanna, Nov 06 2015]:

(1) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n * (A(2*x)/2)^n / n!.

(2) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1) * (A(2*x)/2)^n / n! ).

Example

G.f.: A(x) = x + x^2 + 4*x^3 + 31*x^4 + 412*x^5 + 9322*x^6 +...
where A(x) results from nested inversions of shifted series:
A(x) = Series_Reversion(x - x*B(x)), where
B(x) = x + 2*x^2 + 16*x^3 + 248*x^4 + 6592*x^5 + 298304*x^6 +...;
B(x) = Series_Reversion(x - 2*x*C(x)), where
C(x) = x + 4*x^2 + 64*x^3 + 1984*x^4 + 105472*x^5 + 9545728*x^6 +...;
C(x) = Series_Reversion(x - 4*x*D(x)), where
D(x) = x + 8*x^2 + 256*x^3 + 15872*x^4 + 1687552*x^5 + 305463296*x^6 +...;
D(x) = Series_Reversion(x - 8*x*E(x)), where
E(x) = x + 16*x^2 + 1024*x^3 + 126976*x^4 + 27000832*x^5 + 9774825472*x^6 +...;
E(x) = Series_Reversion(x - 16*x*F(x)), where
F(x) = x + 32*x^2 + 4096*x^3 + 1015808*x^4 + 432013312*x^5 + 312794415104*x^6 +...;
F(x) = Series_Reversion(x - 32*x*G(x)), where
G(x) = x + 64*x^2 + 16384*x^3 + 8126464*x^4 + 6912212992*x^5 + 10009421283328*x^6 +...; ...

Prog

(PARI) {a(n) = my(A=x); for(i=1, n, A = serreverse(x - x*subst(A, x, 2*x)/2 +x*O(x^n)); ); polcoeff(A, n)}
for(n=1, 20, print1(a(n), ", "))
(PARI) {a(n) = my(G=x+x^2); for(k=0, n, G=serreverse(x-2^(n-k)*x*G+x*O(x^n))); polcoeff(G, n)}
for(n=1, 20, print1(a(n), ", "))

Crossrefs

Cf. A194956, A195194, A195196.

Sequence in context: A211194 A237581 A319074 * A141827 A262529 A350608

Adjacent sequences: A195192 A195193 A195194 * A195196 A195197 A195198

Keyword

nonn

Author

Paul D. Hanna, Sep 11 2011

Extensions

Name change by Paul D. Hanna, Nov 06 2015

Status

approved