%I #11 Jan 08 2024 01:44:53
%S 1,1,2,4,10,50,220,1240,6140,32860,602200,5668400,62030200,522328600,
%T 4487190800,62591332000,715163146000,30496564010000,482341877812000,
%U 8342949421288000,124613700640580000,1733826182453140000,36635355834463000000,597186420007933040000
%N Expansion of e.g.f.: exp( Sum_{n>=0} x^(n^2+1)/(n^2+1) ).
%C Conjecture: p | a(n) for n>=p when p is a prime of the form m^2+1 (A002496).
%H G. C. Greubel, <a href="/A214724/b214724.txt">Table of n, a(n) for n = 0..449</a>
%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 4*x^3/3! + 10*x^4/4! + 50*x^5/5! + 220*x^6/6! +...
%e where, by definition,
%e log(A(x)) = x + x^2/2 + x^5/5 + x^10/10 + x^17/17 + x^26/26 + x^37/37 +...
%t With[{m=30}, CoefficientList[Series[Exp[Sum[x^(n^2+1)/(1+n^2), {n,0,m+ 2}]], {x,0,m}], x]*Range[0,m]!] (* _G. C. Greubel_, Jan 07 2024 *)
%o (PARI) {a(n)=n!*polcoeff(exp(sum(k=0,n,x^(k^2+1)/(k^2+1) + x*O(x^n))),n)}
%o for(n=0,21,print1(a(n),", "))
%o (Magma)
%o m:=30;
%o R<x>:=PowerSeriesRing(Rationals(), m+1);
%o Coefficients(R!(Laplace( Exp((&+[x^(n^2+1)/(n^2+1): n in [0..m+2]])) ))); // _G. C. Greubel_, Jan 07 2024
%o (SageMath)
%o m=30
%o def A214724_list(prec):
%o P.<x> = PowerSeriesRing(QQ, prec)
%o return P( exp(sum(x^(n^2+1)/(n^2+1) for n in range(m+3))) ).egf_to_ogf().list()
%o A214724_list(m) # _G. C. Greubel_, Jan 07 2024
%Y Cf. A002496.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jul 26 2012