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A195195 G.f. A(x) satisfies: A( x - x*A(2*x)/2 ) = x. 3

%I

%S 1,1,4,31,412,9322,370792,26998435,3721060012,992975985526,

%T 520211594819608,539421251144944822,1112261291753364074008,

%U 4572473903615507238126100,37530785839971207229105541584,615542030204664293118535562855563,20181157288253257322062384770648622156

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

%H Paul D. Hanna, <a href="/A195195/b195195.txt">Table of n, a(n) for n = 1..50</a>

%F 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).

%F G.f. A(x) also satisfies [from _Paul D. Hanna_, Nov 06 2015]:

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

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

%e G.f.: A(x) = x + x^2 + 4*x^3 + 31*x^4 + 412*x^5 + 9322*x^6 +...

%e where A(x) results from nested inversions of shifted series:

%e A(x) = Series_Reversion(x - x*B(x)), where

%e B(x) = x + 2*x^2 + 16*x^3 + 248*x^4 + 6592*x^5 + 298304*x^6 +...;

%e B(x) = Series_Reversion(x - 2*x*C(x)), where

%e C(x) = x + 4*x^2 + 64*x^3 + 1984*x^4 + 105472*x^5 + 9545728*x^6 +...;

%e C(x) = Series_Reversion(x - 4*x*D(x)), where

%e D(x) = x + 8*x^2 + 256*x^3 + 15872*x^4 + 1687552*x^5 + 305463296*x^6 +...;

%e D(x) = Series_Reversion(x - 8*x*E(x)), where

%e E(x) = x + 16*x^2 + 1024*x^3 + 126976*x^4 + 27000832*x^5 + 9774825472*x^6 +...;

%e E(x) = Series_Reversion(x - 16*x*F(x)), where

%e F(x) = x + 32*x^2 + 4096*x^3 + 1015808*x^4 + 432013312*x^5 + 312794415104*x^6 +...;

%e F(x) = Series_Reversion(x - 32*x*G(x)), where

%e G(x) = x + 64*x^2 + 16384*x^3 + 8126464*x^4 + 6912212992*x^5 + 10009421283328*x^6 +...; ...

%o (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)}

%o for(n=1,20,print1(a(n),", "))

%o (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)}

%o for(n=1,20,print1(a(n),", "))

%Y Cf. A194956, A195194, A195196.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Sep 11 2011

%E Name change by _Paul D. Hanna_, Nov 06 2015

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Last modified August 11 03:41 EDT 2022. Contains 356046 sequences. (Running on oeis4.)