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%I #7 Jul 03 2015 10:16:59
%S 1,2,15,164,2190,33384,561659,10226376,198975366,4101249990,
%T 88985266436,2022670569000,47986654728506,1184722493746988,
%U 30364559922967455,806313807163378768,22146014022165507644,628220131284285896472,18382404744008384580629,554214116675011187495440
%N G.f. A(x) satisfies: A(x) = Series_Reversion( x - x^2*A(x) - x*Integral 2*A(x) dx ).
%F G.f. A(x) satisfies:
%F (1) A(x) = Series_Reversion( x - Sum_{n>=1} (n+1)/n * a(n) * x^(2*n+1) ).
%F (2) A(x) = x + Sum_{n>=1} (n+1)/n * a(n) * A(x)^(2*n+1).
%F Let B(x) = Integral 2*A(x) dx, then
%F (3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (x*A(x) + B(x))^n * x^n / n!.
%F (4) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (x*A(x) + B(x))^n * x^(n-1) / n! ).
%F a(n)/n = A259609(n) for n>=1.
%e G.f.: A(x) = x + 2*x^3 + 15*x^5 + 164*x^7 + 2190*x^9 + 33384*x^11 +...
%e Let B(x) = Integral 2*A(x) dx
%e B(x) = x^2 + x^4 + 5*x^6 + 41*x^8 + 438*x^10 + 5564*x^12 + 80237*x^14 + 1278297*x^16 + 22108374*x^18 +...+ A259609(n)*x^(2*n) +...
%e such that A(x - x^2*A(x) - x*B(x)) = x.
%e Also,
%e A(x) = x + (x*A(x) + B(x))*x + [d/dx (x*A(x) + B(x))^2*x^2]/2! + [d^2/dx^2 (x*A(x) + B(x))^3*x^3]/3! + [d^3/dx^3 (x*A(x) + B(x))^4*x^4]/4! + [d^4/dx^4 (x*A(x) + B(x))^5*x^5]/5! +...
%e Logarithmic series:
%e log(A(x)/x) = (x*A(x) + B(x)) + [d/dx (2*x*A(x) + B(x))^2*x]/2! + [d^2/dx^2 (2*x*A(x) + B(x))^3*x^2]/3! + [d^3/dx^3 (2*x*A(x) + B(x))^4*x^3]/4! + [d^4/dx^4 (2*x*A(x) + B(x))^5*x^4]/5! +...
%o (PARI) {a(n)=local(A=x); for(i=0, n, A = serreverse(x - x^2*A - x*intformal(2*A) +x*O(x^(2*n)))); polcoeff(A, 2*n-1)}
%o for(n=1, 25, print1(a(n), ", "))
%Y Cf. A259609.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jun 30 2015
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