The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A318634 a(n) = coefficient of x^(2*n-1)*y^(2*n)/(2*n-1)! in Log( Sum_{n>=0} (n^2 + y^2)^n * x^n/n! ), for n>=1. 6

%I #16 Mar 20 2024 09:18:37

%S 1,6,480,122640,66044160,61482516480,88135315107840,

%T 180378921026304000,499734635092800307200,1801642618822079338905600,

%U 8199046303785011864744755200,45976521975711536997953490124800,311502479360401852390993821696000000,2508845886467091418046335123571343360000,23693183471722887844366765687378500648960000

%N a(n) = coefficient of x^(2*n-1)*y^(2*n)/(2*n-1)! in Log( Sum_{n>=0} (n^2 + y^2)^n * x^n/n! ), for n>=1.

%C E.g.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1)/(2*n-1)! equals the logarithm of the e.g.f. of A318633.

%H Vaclav Kotesovec, <a href="/A318634/b318634.txt">Table of n, a(n) for n = 1..145</a> (terms 1..75 from Paul D. Hanna)

%F a(n) ~ 5^(-1/4) * 2^(3*n - 3/2) * (1 + sqrt(5))^(n - 3/2) * exp((1 - sqrt(5))*n + (sqrt(5) - 3)/2) * n^(2*n-3). - _Vaclav Kotesovec_, Mar 20 2024

%e E.g.f.: 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! + ...

%e The e.g.f. A(x) may also be written using somewhat reduced coefficients

%e A(x) = x + x^3 + 8*x^5/2! + 146*x^7/3! + 4368*x^9/4! + 184832*x^11/5! + 10190656*x^13/6! + 695211120*x^15/7! + 56648897024*x^17/8! + 5374487515904*x^19/9! + ... + a(n)*(n-1)!/(2*n-1)! * x^(2*n-1)/(n-1)! + ...

%e Exponentiation yields the e.g.f. of A318633:

%e exp(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! +...+ A318633(n)*x^n/n! + ...

%e which equals

%e Limit_{N->oo} [ Sum_{n>=0} (N^2 + n^2)^n * (x/N)^n/n! ]^(1/N).

%o (PARI) {a(n) = (2*n-1)! * polcoeff( polcoeff( log( sum(m=0, 2*n, (m^2 + y^2)^m *x^m/m! ) +x*O(x^(2*n)) ), 2*n-1, x), 2*n, y)}

%o for(n=1, 20, print1(a(n), ", "))

%Y Cf. A318633, A266526, A319834.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Sep 04 2018

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified August 7 06:05 EDT 2024. Contains 375008 sequences. (Running on oeis4.)