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%I #13 Mar 21 2024 05:43:56
%S 1,1,1,7,25,541,3361,135451,1179697,72062425,800549281,66556219951,
%T 895033831561,94883671398517,1494100965599425,193406329375101571,
%U 3484870135007879521,534176530923146533681,10828212525851495841217,1921147013664776094354775,43243444110218718242048761,8725818848745836121239731021,215849236633217233817102676961,48851798675231712377778383815147
%N E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N^2 + n^2)^n * (x/N)^n/n! ]^(1/N).
%C Compare to:
%C (C1) exp(x) = Limit_{N->oo} [ Sum_{n>=0} (N^2 + n)^n * (x/N)^n/n! ]^(1/N).
%C (C2) W(x) = Limit_{N->oo} [ Sum_{n>=0} (N^2 + N*n)^n * (x/N)^n/n! ]^(1/N), where W(x) = LambertW(-x)/(-x).
%H Paul D. Hanna, <a href="/A318633/b318633.txt">Table of n, a(n) for n = 0..150</a>
%F E.g.f. exp( Sum_{n>=0} A318634(n)*x^n/n! ), where A318634(n) = [x^n*y^(n+1)/n!] log( Sum_{n>=0} (n^2 + y^2)^n *x^n/n! ).
%F a(n) ~ c * 2^(3*n/2) * (1 + sqrt(5))^(n/2) * exp((3 - sqrt(5))*n/2) * n^(n-2) / (2^(n-2) * exp(n)), where c = 0.06299092342072472381892574307... if n is even and c = 0.21603924353200238368086490887... if n is odd. - _Vaclav Kotesovec_, Mar 21 2024
%e E.g.f.: A(x) = 1 + x + x^2/2! + 7*x^3/3! + 25*x^4/4! + 541*x^5/5! + 3361*x^6/6! + 135451*x^7/7! + 1179697*x^8/8! + 72062425*x^9/9! + 800549281*x^10/10! + ...
%e where A(x) equals the limit, as N -> oo, of the series
%e [1 + (N^2+1)*(x/N) + (N^2+2^2)^2*(x/N)^2/2! + (N^2+3^2)^3*(x/N)^3/3! + (N^2+4^2)^4*(x/N)^4/4! + (N^2+5^2)^5*(x/N)^5/5! + (N^2+6^2)^6*(x/N)^6/6! +...]^(1/N).
%e RELATED SERIES.
%e (a) The following limit exists:
%e G(x) = Limit_{N->oo} [ Sum_{n>=0} (N^2 + n^2)^n * (x/N)^n/n! ] / A(x)^N
%e where G(x) is an even function that begins
%e G(x) = 1 + 6*x^2/2! + 636*x^4/4! + 200520*x^6/6! + 127426320*x^8/8! + 135927439200*x^10/10! + 218711489895360*x^12/12! + 494824278347118720*x^14/14! 1497827553156635500800*x^16/16! +...
%e (b) The logarithm of the e.g.f. A(x) is an odd function that begins:
%e Log(A(x)) = x + 6*x^3/3! + 480*x^5/5! + 122640*x^7/7! + 66044160*x^9/9! + 61482516480*x^11/11! + 88135315107840*x^13/13! + 180378921026304000*x^15/15! + ... + A318634(n)*x^(2*n-1)/(2*n-1)! +...
%o (PARI) {A318634(n) = n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m^2 + y^2)^m *x^m/m! ) +x*O(x^n) ), n, x), n+1, y)}
%o {a(n) = n! * polcoeff( exp( sum(m=1, n+1, A318634(m)*x^m/m! ) +x*O(x^n)), n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) /* Informal listing of terms 0..30 */
%o \p100
%o P(n) = sum(k=0, 31, (n^2+k^2)^(k) * x^k/k! +O(x^31))
%o Vec(round( serlaplace( subst(P(10^100)^(1/10^100), x, x/10^100) )*1.) )
%Y Cf. A318634, A266481.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Sep 04 2018
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