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%I #13 Sep 30 2018 07:24:23
%S 1,2,22,432,12220,451480,20591784,1117635008,70348179472,
%T 5037843612960,404453425948000,35977638091065088,3512312454013520832,
%U 373346162796913784192,42922941487808176036480,5307003951337894697856000,702183042248318469458657536,98997224309112273722486891008,14815674464782854979394204308992,2345767767928443601985964232355840,391750020994050554579656281189760000,68820978855281989513379320801711429632
%N E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(2*n) * (x/N)^n/n! ] / F(x)^N, where F(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(2*n) * (x/N)^n/n! ]^(1/N).
%C The e.g.f. A(x) of this sequence also satisfies:
%C A(x*y) = Limit_{N->oo} [ Sum_{n>=0} (N + n*y)^(2*n) * (x/N)^n/n! ] / G(x,y)^N
%C where
%C G(x,y) = Limit_{N->oo} [ Sum_{n>=0} (N + n*y)^(2*n) * (x/N)^n/n! ]^(1/N)
%C for all real y.
%F E.g.f.: exp( Sum_{n>=1} A266521(n,n)*x^n/n! ), where the e.g.f. of triangle A266521 is Log(Sum_{n>=0} (n + y)^(2*n) * x^n/n!). - _Paul D. Hanna_, Sep 30 2018
%e E.g.f.: A(x) = 1 + 2*x + 22*x^2/2! + 432*x^3/3! + 12220*x^4/4! + 451480*x^5/5! + 20591784*x^6/6! + 1117635008*x^7/7! + 70348179472*x^8/8! + 5037843612960*x^9/9! + 404453425948000*x^10/10! + ...
%e such that
%e A(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(2*n) * (x/N)^n/n! ] / F(x)^N
%e where
%e F(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(2*n) * (x/N)^n/n! ]^(1/N)
%e and
%e F(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! + 7022842421301*x^10/10! +...+ A266481(n)*x^n/n! + ...
%e RELATED SERIES.
%e log(A(x)) = 2*x + 18*x^2/2! + 316*x^3/3! + 8272*x^4/4! + 288048*x^5/5! + 12523584*x^6/6! + 652959872*x^7/7! + 39701769216*x^8/8! + 2758053332736*x^9/9! + ... + A266521(n,n)*x^n/n! + ...
%Y Cf. A266481, A266523, A266524, A266525, A266521.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 30 2015
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