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a(n) = coefficient of x^n*y^(n+1)/n! in Log( Sum_{n>=0} (n + y)^(2*n) * x^n/n! ), for n>=1.
6

%I #9 Mar 20 2024 09:19:06

%S 1,4,42,752,19360,654912,27546736,1388207872,81621893376,

%T 5488951731200,415721105434624,35026876903256064,3250356630453317632,

%U 329437813126362185728,36214170617862339840000,4291812357982293898231808,545518054282041342531076096,74032137722410904128877494272,10684317262536125210489796296704,1634019721630446295055397683200000

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

%C Equals the logarithm of the e.g.f. of A266481.

%C Equals the right border of triangle A266521.

%H Vaclav Kotesovec, <a href="/A266526/b266526.txt">Table of n, a(n) for n = 1..230</a>

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

%e E.g.f: A(x) = x + 4*x^2/2! + 42*x^3/3! + 752*x^4/4! + 19360*x^5/5! + 654912*x^6/6! + 27546736*x^7/7! + 1388207872*x^8/8! + 81621893376*x^9/9! + 5488951731200*x^10/10! +...

%e where exponentiation yields the e.g.f. of A266481:

%e exp(A(x)) = 1 + x + 5*x^2/2! + 55*x^3/3! + 993*x^4/4! + 25501*x^5/5! + 857773*x^6/6! + 35850795*x^7/7! + 1795564865*x^8/8! + 104972371417*x^9/9! +...+ A266481(n)*x^n/n! +...

%e which equals

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

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

%o for(n=1,30, print1(a(n),", "))

%Y Cf. A266521, A266481.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jan 01 2016

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Last modified September 20 06:55 EDT 2024. Contains 376067 sequences. (Running on oeis4.)