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A259272 L.g.f.: log(G(x)/x) where G(x) is the g.f. of A259270. 3

%I

%S 4,64,1264,28064,675504,17304544,466669536,13155395904,385761948592,

%T 11725112136544,368418702111904,11942661856743104,398739401015768544,

%U 13694120483326491328,483240865665765964224,17505677353238744717952,650483549418017027126704,24776136706182576128200288

%N L.g.f.: log(G(x)/x) where G(x) is the g.f. of A259270.

%C G.f. G(x) of A259270 satisfies: G( x - 2*G(x)*H(x) ) = x, where H'(x) = 2*G(x).

%H Paul D. Hanna, <a href="/A259272/b259272.txt">Table of n, a(n) for n = 1..199</a>

%e L.g.f.: L(x) = 4*x^2/2 + 64*x^4/4 + 1264*x^6/6 + 28064*x^8/8 + 675504*x^10/10 +...

%e where the g.f. of A259270 begins:

%e G(x) = x*exp(L(x)) = x + 2*x^3 + 18*x^5 + 244*x^7 + 4090*x^9 + 78636*x^11 +...+ A259270(n)*x^(2*n-1) +...

%e Now let H(x) = Integral G(x) dx, then

%e L(x) = 2*G(x)*H(x)/x + [d/dx 4*G(x)^2*H(x)^2/x]/2! + [d^2/dx^2 8*G(x)^3*H(x)^3/x]/3! + [d^3/dx^3 16*G(x)^4*H(x)^4/x]/4! + [d^4/dx^4 32*G(x)^5*H(x)^5/x]/5! +...

%o (PARI) {a(n)=local(A=x+x*O(x^n), B=x^2); for(i=1, n, B=intformal(2*A); A = serreverse(x - 2*A*B +O(x^(2*n+2)))); 2*n*polcoeff(log(A/x), 2*n)}

%o for(n=1, 25, print1(a(n), ", "))

%Y Cf. A259270, A259271.

%K nonn

%O 1,1

%A _Paul D. Hanna_, Jun 29 2015

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Last modified September 19 06:16 EDT 2020. Contains 337177 sequences. (Running on oeis4.)