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%I #21 Jun 30 2015 17:30:38
%S 1,1,6,61,818,13106,238636,4796157,104441690,2433287430,60109378452,
%T 1563967551762,42642719385012,1213585435256772,35935842038596312,
%U 1104324433869399581,35143747323887055722,1156109729255078573566,39253565467948968047876,1373742020268961592289798
%N G.f. A(x) satisfies: A'(x) = 2 * Series_Reversion( x - A(x)*A'(x) ).
%C G.f. G(x) of A259270 satisfies: G(x) = Series_Reversion( x - 2*A(x)*G(x) ) such that G(x) = A'(x)/2, where A(x) = Sum_{n>=1} a(n)*x^(2*n) is the g.f. of this sequence.
%H Paul D. Hanna, <a href="/A259271/b259271.txt">Table of n, a(n) for n = 1..200</a>
%F G.f. A(x) satisfies:
%F (1) A'(x) = 2*x + 2*Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^n * A'(x)^n / n!.
%F (2) A'(x) = 2*x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^n * A'(x)^n / (n!*x) ).
%F a(n) = A259270(n) / n.
%F a(n) == 1 (mod 2) iff n is a power of 2 (conjecture).
%e G.f.: A(x) = x^2 + x^4 + 6*x^6 + 61*x^8 + 818*x^10 + 13106*x^12 + 238636*x^14 +...
%e Let G(x) be the g.f. of A259270 such that
%e G(x) = A'(x)/2 = x + 2*x^3 + 18*x^5 + 244*x^7 + 4090*x^9 + 78636*x^11 + 1670452*x^13 + 38369256*x^15 +...+ A259270(n)*x^(2*n-1) +...
%e then G( x - 2*A(x)*G(x) ) = x.
%e Also,
%e A'(x)/2 = x + A(x)*A'(x) + [d/dx A(x)^2*A'(x)^2]/2! + [d^2/dx^2 A(x)^3*A'(x)^3]/3! + [d^3/dx^3 A(x)^4*A'(x)^4]/4! + [d^4/dx^4 A(x)^5*A'(x)^5]/5! +...
%o (PARI) {a(n)=local(A=x^2); for(i=1, n, A=intformal(2*serreverse(x - A*A' +O(x^(2*n))))); polcoeff(A, 2*n)}
%o for(n=1, 25, print1(a(n), ", "))
%o (PARI) {a(n)=local(A,G=x+x*O(x^n)); for(i=1, n, A=intformal(2*G); G = serreverse(x - 2*A*G +O(x^(2*n)))); polcoeff(A, 2*n)}
%o for(n=1, 25, print1(a(n), ", "))
%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o {a(n)=local(A=x^2); for(i=1, n, A = 2*intformal(x + sum(m=1, n+1, Dx(m-1, A^m*(A')^m/m!)) +O(x^(2*n+1)))); polcoeff(A, 2*n)}
%o for(n=1, 25, print1(a(n), ", "))
%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o {a(n)=local(A=x^2); for(i=1, n, A = 2*intformal(x*exp(sum(m=1, n, Dx(m-1, A^m*(A')^m/(m!*x))) +O(x^(2*n+1))))); polcoeff(A, 2*n)}
%o for(n=1, 25, print1(a(n), ", "))
%Y Cf. A259270, A259272, A259269.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Jun 29 2015
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