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 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
 1, 6, 480, 122640, 66044160, 61482516480, 88135315107840, 180378921026304000, 499734635092800307200, 1801642618822079338905600, 8199046303785011864744755200, 45976521975711536997953490124800, 311502479360401852390993821696000000, 2508845886467091418046335123571343360000, 23693183471722887844366765687378500648960000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. LINKS Vaclav Kotesovec, Table of n, a(n) for n = 1..145 (terms 1..75 from Paul D. Hanna) FORMULA 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 EXAMPLE 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! + ... The e.g.f. A(x) may also be written using somewhat reduced coefficients 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)! + ... Exponentiation yields the e.g.f. of A318633: 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! + ... which equals Limit_{N->oo} [ Sum_{n>=0} (N^2 + n^2)^n * (x/N)^n/n! ]^(1/N). PROG (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)} for(n=1, 20, print1(a(n), ", ")) CROSSREFS Cf. A318633, A266526, A319834. Sequence in context: A084108 A269761 A229503 * A006713 A248362 A006712 Adjacent sequences: A318631 A318632 A318633 * A318635 A318636 A318637 KEYWORD nonn AUTHOR Paul D. Hanna, Sep 04 2018 STATUS approved

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Last modified July 20 23:17 EDT 2024. Contains 374461 sequences. (Running on oeis4.)