A298697 - OEIS

%I #14 Feb 09 2018 03:26:07

%S 1,1,3,49,2461,234921,35329471,7619812873,2224009220889,

%T 843836219043121,403933464464241691,238294863084819823041,

%U 169987570007123851643893,144326641180625057628760729,143904971264557527854144891031,166568515940712946346074697276761,221584133833606888317064501087446961

%N E.g.f. A(x) satisfies: 1 = Sum_{n>=0} (n+1)^(2*(n-1)) * x^n/n! / A(x)^((n+1)^2).

%C Compare e.g.f. to: 1 = Sum_{n>=0} m * (m*(n+1))^(n-1) * x^n/n! / exp(x)^(m*(n+1)) holds for fixed m.

%H Paul D. Hanna, <a href="/A298697/b298697.txt">Table of n, a(n) for n = 0..120</a>

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 49*x^3/3! + 2461*x^4/4! + 234921*x^5/5! + 35329471*x^6/6! + 7619812873*x^7/7! + 2224009220889*x^8/8! + ...

%e such that

%e 1 = 1/A(x) + x/A(x)^4 + 3^2*x^2/2!/A(x)^9 + 4^4*x^3/3!/A(x)^16 + 5^6*x^4/4!/A(x)^25 + 6^8*x^5/5!/A(x)^36 + 7^10*x^6/6!/A(x)^49 + ...

%e RELATED SERIES.

%e As a power series in x with reduced fractional coefficients,

%e A(x) = 1 + x + 3/2*x^2 + 49/6*x^3 + 2461/24*x^4 + 78307/40*x^5 + 35329471/720*x^6 + 7619812873/5040*x^7 + 741336406963/13440*x^8 + 843836219043121/362880*x^9 + 57704780637748813/518400*x^10 + ...

%e The logarithm of e.g.f. A(x) begins:

%e log(A(x)) = x + 2*x^2/2! + 42*x^3/3! + 2268*x^4/4! + 222240*x^5/5! + 33870120*x^6/6! + 7364669760*x^7/7! + 2161424593440*x^8/8! + 823377654610560*x^9/9! + 395340117541407360*x^10/10! ...

%t terms = 17; A[_] = 1; Do[A[x_] = A[x] - 1 + Sum[(n + 1)^(2*(n - 1))*x^n/n!/A[x]^((n+1)^2), {n, 0, terms}] + O[x]^terms, {terms}]; CoefficientList[ A[x], x]*Range[0, terms-1]! (* _Jean-François Alcover_, Feb 09 2018 *)

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0); A[#A] = Vec(sum(m=0,#A, ((m+1)^2)^(m-1) * x^m/m! /Ser(A)^((m+1)^2) ))[#A] ); n!*A[n+1]}

%o for(n=0,20,print1(a(n),", "))

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 08 2018