A300047 - OEIS

%I #3 Feb 25 2018 08:12:26

%S 1,1,8,270,35472,18318288,37611139104,308338698386160,

%T 10105807430398162176,1324669305373789482964224,

%U 694520145536868530329362481152,1456521257891915020152240334073326080,12218201898131114878545053215635303915614208,409974971372215896118360380056730403849666983370752

%N E.g.f. L(x) satisfies: L(x) = log(1 + Integral exp( L(4*x)/2 ) dx).

%F E.g.f.: L(x) = log(G(x)) where G(x) is the e.g.f. of A300045.

%e E.g.f.: L(x) = x + x^2/2! + 8*x^3/3! + 270*x^4/4! + 35472*x^5/5! + 18318288*x^6/6! + 37611139104*x^7/7! + 308338698386160*x^8/8! + 10105807430398162176*x^9/9! + ...

%e Related series.

%e exp(L(x)) = 1 + x + 2*x^2/2! + 12*x^3/3! + 312*x^4/4! + 37008*x^5/5! + 18540576*x^6/6! + ... + A300045(n)*x^n/n! + ...

%e exp(L(4*x)/2) = 1 + 2*x + 12*x^2/2! + 312*x^3/3! + 37008*x^4/4! + 18540576*x^5/5! + ... + A300045(n+1)*x^n/n! + ...

%e exp(L(2*x)/2) = 1 + x + 3*x^2/2! + 39*x^3/3! + 2313*x^4/4! + 579393*x^5/5! + 589702779*x^6/6! + ... + A300046(n)*x^n/n! + ...

%o (PARI) {a(n) = my(A=1+x); for(i=1,n, A = 1 + intformal(subst(A,x,4*x)^(1/2) +x*O(x^n) )); n!*polcoeff(log(A),n)}

%o for(n=1,16, print1(a(n),", "))

%Y Cf. A300045, A300046.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Feb 25 2018