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%I #6 May 09 2016 18:12:13
%S 1,1,1,1,2,7,23,65,167,418,1078,2927,8261,23677,67917,194401,557097,
%T 1604805,4655373,13592002,39875297,117379888,346445156,1025077295,
%U 3040774879,9043131251,26958950829,80545589421,241122613367,723125785251,2172271809579,6535785964961,19693769446456,59425319779021,179551868445081,543187254954865,1645200904479900
%N G.f. A(x) satisfies: A( A(x)^2 - A(x)^4 ) = x*A(x) + x*A(x)^2.
%H Paul D. Hanna, <a href="/A272819/b272819.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. A(x) satisfies: A( A(x^2 - x^4) / (x + x^2) ) = x.
%e G.f.: A(x) = x + x^2 + x^3 + x^4 + 2*x^5 + 7*x^6 + 23*x^7 + 65*x^8 + 167*x^9 + 418*x^10 + 1078*x^11 + 2927*x^12 +...
%e where A( A(x)^2 - A(x)^4 ) = x*A(x) + x*A(x)^2.
%e RELATED SERIES.
%e A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 7*x^6 + 20*x^7 + 65*x^8 + 194*x^9 + 528*x^10 + 1374*x^11 + 3597*x^12 + 9762*x^13 + 27475*x^14 +...
%e A(x)^4 = x^4 + 4*x^5 + 10*x^6 + 20*x^7 + 39*x^8 + 92*x^9 + 268*x^10 + 824*x^11 + 2431*x^12 + 6824*x^13 + 18720*x^14 + + 51696*x^15 +...
%e A(x)^2 - A(x)^4 = x^2 + 2*x^3 + 2*x^4 - 3*x^6 + 26*x^8 + 102*x^9 + 260*x^10 + 550*x^11 + 1166*x^12 + 2938*x^13 + 8755*x^14 + 27190*x^15 +...
%e A(x) + A(x)^2 = x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 14*x^6 + 43*x^7 + 130*x^8 + 361*x^9 + 946*x^10 + 2452*x^11 + 6524*x^12 + 18023*x^13 +...
%o (PARI) {a(n) = my(A=x+x^2); for(i=1,n, A = serreverse( subst(A,x, x^2-x^4 +x^2*O(x^n)) / (x+x^2) ) );polcoeff(A,n)}
%o for(n=1,40,print1(a(n),", "))
%K nonn
%O 1,5
%A _Paul D. Hanna_, May 09 2016
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