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A109777 G.f. = f(x), where f(x)^2 = o.g.f. for A088313 (with offset 0).

%I A109777 #13 Sep 03 2024 08:24:41
%S A109777 1,1,3,15,101,829,7891,84735,1009065,13170841,186798003,2859068831,
%T A109777 46960097413,823787983021,15370572776091,303929827526887,
%U A109777 6348320745774993,139663855708967665,3227812335094695171,78180132507785056399,1980181972528939129861,52344600987011191983613
%N A109777 G.f. = f(x), where f(x)^2 = o.g.f. for A088313 (with offset 0).
%H A109777 N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.NT/0509316">On the Integrality of n-th Roots of Generating Functions</a>, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%e A109777 The present sequence has g.f. f(x) = 1 + x + 3*x^2 + 15*x^3 + ...
%e A109777 A088313 [1,2,7,36,242,...] has e.g.f. = sinh(x/(1-x)) = x + x^2 + 7/6*x^3 + 3/2*x^4 + 241/120*x^5 + 65/24*x^6 + 18271/5040*x^7 + ... and (with offset 0) o.g.f. = 1 + 2*x^2 + 7*x^3 + 36*x^4 + ... = f(x)^2.
%t A109777 nmax = 22;
%t A109777 f[x_] = Sqrt[Sum[SeriesCoefficient[Sinh[x/(1-x)], {x, 0, n}] n! x^n, {n, 0, nmax}]] + O[x]^nmax // Normal;
%t A109777 List @@ f[x] /. x -> 1 (* _Jean-François Alcover_, Oct 08 2018 *)
%K A109777 nonn
%O A109777 0,3
%A A109777 _N. J. A. Sloane_ and _Nadia Heninger_, Aug 15 2005