A301386 - OEIS

%I #8 Oct 07 2020 02:54:28

%S 1,1,5,49,745,16001,472621,19659025,1211940689,112956505345,

%T 15173515994581,2723523227716721,615752085911442745,

%U 169058306483325138049,55091534512604633032445,20970346191311260824482641,9209286575036784425924388001,4618748590360572876528407678465,2622629436345978251926864051089829,1673388831386832236982492861374330545,1191875845170136556300137233780466470281

%N E.g.f. A(x) satisfies: [x^n] A(x)^(-n) = (2*n - 3) * [x^(n-1)] A(x)^(-n) for n>=1.

%H Paul D. Hanna, <a href="/A301386/b301386.txt">Table of n, a(n) for n = 0..200</a>

%F E.g.f. A(x) satisfies: A(x) = exp( x * (A(x) + 3*x*A'(x)) / (A(x) + x*A'(x)) ).

%F a(n) ~ c * 2^n * n!^2 / n^2, where c = 0.061831520522917842264623558587952196792... - _Vaclav Kotesovec_, Oct 07 2020

%e E.g.f.: A(x) = 1 + x + 5*x^2/2! + 49*x^3/3! + 745*x^4/4! + 16001*x^5/5! + 472621*x^6/6! + 19659025*x^7/7! + 1211940689*x^8/8! + 112956505345*x^9/9! + ...

%e The table of coefficients in A(x)^(-n) begins:

%e n=1: [(1), (-1), -3/2, -25/6, -359/24, -2587/40, -245099/720, ...];

%e n=2: [1, (-2), (-2), -16/3, -58/3, -1304/15, -22016/45, ...];

%e n=3: [1, -3, (-3/2), (-9/2), -141/8, -3441/40, -42579/80, ...];

%e n=4: [1, -4, 0, (-8/3), (-40/3), -376/5, -23624/45, -293096/63, ...];

%e n=5: [1, -5, 5/2, -5/6, (-215/24), (-1505/24), -72055/144, ...];

%e n=6: [1, -6, 6, 0, -6, (-264/5), (-2376/5), -173388/35, ...];

%e n=7: [1, -7, 21/2, -7/6, -119/24, -1869/40, (-327971/720), (-3607681/720), ...]; ...

%e in which the coefficients in parenthesis are related by

%e -1 = -1*(1); -2 = 1*(-2); -9/2 = 3*(3/2); -40/3 = 5*(-8/3); -1505/24 = 7*(-215/24); -2376/5 = 9*(-264/5); -3607681/720 = 11*(-327971/720); ...

%e illustrating: [x^n] A(x)^(-n) = (2*n - 3) * [x^(n-1)] A(x)^(-n).

%e LOGARITHMIC PROPERTY.

%e The logarithm of the e.g.f. is an integer power series in x satisfying

%e log(A(x)) = x * (1 + 3*x*A'(x)/A(x)) / (1 + x*A'(x)/A(x));

%e explicitly,

%e log(A(x)) = x + 2*x^2 + 6*x^3 + 22*x^4 + 94*x^5 + 474*x^6 + 2974*x^7 + 24630*x^8 + 271710*x^9 + 3799570*x^10 + ... + A301385(n)*x^n + ...

%o (PARI) {a(n) = my(A=1); for(i=1, n, A = exp( x*(A + 3*x*A')/(A + x*A' +x*O(x^n)) ) ); n!*polcoeff(A, n)}

%o for(n=0, 25, print1(a(n), ", "))

%Y Cf. A301385, A300735, A300986.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Mar 20 2018