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A319147
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E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N^2 + N*n + n^2)^n * (x/N)^n/n! ]^(1/N).
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6
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1, 1, 3, 22, 269, 4776, 111967, 3280264, 115550073, 4762181440, 224954474651, 11987717900544, 711604917300037, 46572971758429312, 3332107859592406455, 258748811312125854976, 21674785904235983431793, 1948303837796264786497536, 187062919027712092164076723, 19107058023481400501276569600
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OFFSET
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0,3
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COMMENTS
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Compare to:
(C1) exp(x) = Limit_{N->oo} [ Sum_{n>=0} (N^2 + n)^n * (x/N)^n/n! ]^(1/N).
(C2) W(x) = Limit_{N->oo} [ Sum_{n>=0} (N^2 + N*n)^n * (x/N)^n/n! ]^(1/N), where W(x) = LambertW(-x)/(-x).
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LINKS
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FORMULA
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E.g.f. exp( Sum_{n>=0} L(n)*x^n/n! ), where L(n) = [x^n*y^(n+1)/n!] log( Sum_{n>=0} (n^2 + n*y + y^2)^n *x^n/n! ).
a(n) ~ c * d^n * n! / n^(5/2), where d = 6.16018341007619464488... and c = 0.240315927519139896... - Vaclav Kotesovec, Mar 19 2022
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 269*x^4/4! + 4776*x^5/5! + 111967*x^6/6! + 3280264*x^7/7! + 115550073*x^8/8! + 4762181440*x^9/9! + 224954474651*x^10/10! + ...
where A(x) equals the limit, as N -> oo, of the series
[1 + (N^2+N+1)*(x/N) + (N^2+2*N+2^2)^2*(x/N)^2/2! + (N^2+3*N+3^2)^3*(x/N)^3/3! + (N^2+4*N+4^2)^4*(x/N)^4/4! + (N^2+5*N+5^2)^5*(x/N)^5/5! + (N^2+6*N+6^2)^6*(x/N)^6/6! +...]^(1/N).
RELATED SERIES.
(a) The logarithm of the g.f. A(x) begins:
log(A(x)) = x + 2*x^2/2! + 15*x^3/3! + 184*x^4/4! + 3325*x^5/5! + 79056*x^6/6! + 2345539*x^7/7! + 83505920*x^8/8! + 3472829721*x^9/9! + ... + A319834(n)*x^n/n! + ...
where A319834(n) = [x^n*y^(n+1)/n!] log( Sum_{n>=0} (n^2 + n*y + y^2)^n * x^n/n! );
that is, the coefficients in the logarithm of e.g.f A(x) equals the coefficients of y^(n+1)*x^n/n! in the series given by
(b) log( Sum_{n>=0} (n^2 + n*y + y^2)^n * x^n/n! ) = (y^2 + y + 1)*x + (2*y^3 + 9*y^2 + 14*y + 15)*x^2/2! + (15*y^4 + 107*y^3 + 366*y^2 + 639*y + 683)*x^3/3! + (184*y^5 + 2038*y^4 + 10432*y^3 + 32308*y^2 + 58720*y + 62038)*x^4/4! + (3325*y^6 + 50469*y^5 + 367155*y^4 + 1636590*y^3 + 4833195*y^2 + 8940045*y + 9342629)*x^5/5! + (79056*y^7 + 1565256*y^6 + 15015936*y^5 + 90978240*y^4 + 376955520*y^3 + 1085556216*y^2 + 2027376336*y + 2100483216)*x^6/6! + ...
(c) The following limit exists
G(x) = Limit_{N->oo} [ Sum_{n>=0} (N^2 + N*n + n^2)^n * (x/N)^n/n! ] / A(x)^N
where
G(x) = 1 + x + 10*x^2/2! + 135*x^3/3! + 2764*x^4/4! + 72665*x^5/5! + 2362896*x^6/6! + 91282975*x^7/7! + 4088186320*x^8/8! + 208223576721*x^9/9! + ...
the logarithm of which begins
log(G(x)) = x + 9*x^2/2! + 107*x^3/3! + 2038*x^4/4! + 50469*x^5/5! + 1565256*x^6/6! + 58095463*x^7/7! + 2513768496*x^8/8! + ... + D(n)*x^n/n! + ...
where D(n) = [x^n*y^n/n!] log( Sum_{n>=0} (n^2 + n*y + y^2)^n * x^n/n! ).
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PROG
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(PARI) {L(n) = n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m^2 + m*y + y^2)^m *x^m/m! ) +x*O(x^n) ), n, x), n+1, y)}
{a(n) = n! * polcoeff( exp( sum(m=1, n+1, L(m)*x^m/m! ) +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* Informal listing of terms 0..30 */
\p100
P(n) = sum(k=0, 31, (n^2 + n*k + k^2)^k * x^k/k! +O(x^31))
Vec(round( serlaplace( subst(P(10^100)^(1/10^100), x, x/10^100) )*1.) )
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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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