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A266522
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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).
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4
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1, 2, 22, 432, 12220, 451480, 20591784, 1117635008, 70348179472, 5037843612960, 404453425948000, 35977638091065088, 3512312454013520832, 373346162796913784192, 42922941487808176036480, 5307003951337894697856000, 702183042248318469458657536, 98997224309112273722486891008, 14815674464782854979394204308992, 2345767767928443601985964232355840, 391750020994050554579656281189760000, 68820978855281989513379320801711429632
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OFFSET
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0,2
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COMMENTS
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The e.g.f. A(x) of this sequence also satisfies:
A(x*y) = Limit_{N->oo} [ Sum_{n>=0} (N + n*y)^(2*n) * (x/N)^n/n! ] / G(x,y)^N
where
G(x,y) = Limit_{N->oo} [ Sum_{n>=0} (N + n*y)^(2*n) * (x/N)^n/n! ]^(1/N)
for all real y.
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LINKS
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FORMULA
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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
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EXAMPLE
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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! + ...
such that
A(x) = 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)
and
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! + ...
RELATED SERIES.
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! + ...
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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