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A359461
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a(n) = coefficient of x^n/n! in A(x) such that A(x) = Sum_{n>=0} x^n * ( (A(x)^sqrt(n) + x)^sqrt(n) + A(x)^n/(1 + x*A(x)^sqrt(n))^sqrt(n) )/2.
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3
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1, 1, 4, 27, 264, 3480, 57960, 1168860, 27716080, 755797392, 23309811000, 802356730560, 30495894175296, 1268569374923136, 57327261461502032, 2796658399257297120, 146484112541333548800, 8199099498574437696000, 488395687438426037605920, 30847715523237047711124096, 2059258090155754103465678080
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OFFSET
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0,3
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COMMENTS
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In general, if A(x) = 1 + Sum_{n>0} a(n) * x^n/n! = 1 + Sum_{n>0} x^n * ( (A(x)^s(n) + x)^s(n) + 1/(1/A(x)^s(n) + x)^s(n) )/2, for some sequence s(n), then a(1) = 1, a(2) = 2! + 2*a(1)*s(1)^2, a(3) = 3! + 3*s(1)^2 + a(1)*(-6*s(1)^2 + 6*s(2)^2) + a(1)^2*(-3*s(1)^2 + 3*s(1)^4) + a(2)*(3*s(1)^2), and so on. - Michael Somos, Jan 02 2023
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LINKS
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FORMULA
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E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! may be defined by the following.
(1) A(x) = Sum_{n>=0} x^n * ( (A(x)^sqrt(n) + x)^sqrt(n) + A(x)^n/(1 + x*A(x)^sqrt(n))^sqrt(n) )/2.
(2) A(x) = Sum_{n>=0} x^n * ( (A(x)^sqrt(n) + x)^sqrt(n) + 1/(1/A(x)^sqrt(n) + x)^sqrt(n) )/2.
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 264*x^4/4! + 3480*x^5/5! + 57960*x^6/6! + 1168860*x^7/7! + 27716080*x^8/8! + 755797392*x^9/9! + 23309811000*x^10/10! + 802356730560*x^11/11! + 30495894175296*x^12/12! + ...
a(0) = 1. a(1) = 1. a(2) = 2 + 2*a(1). a(3) = 9 + 6*a(1) + 3*a(2). a(4) = 36 + 60*a(1) + 12*a(1)^2 + 12*a(2) + a(3). - Michael Somos, Jan 02 2023
SPECIFIC VALUES.
A(x) = 2 at x = 0.27668637086273541040814473225204080161288734095023351327332...
A(1/4) = 1.60113273405540559575215387233185272854944797616482632040783...
A(1/5) = 1.35563819301497088298535445664511164436786966959434923908625...
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PROG
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(PARI) /* must set precision suitable for desired number of terms */
\p200
{a(n) = my(A=1+x); for(i=1, 30, A = sum(m=0, n, x^m/2 * ( ((A +x*O(x^n))^sqrt(m) + x)^sqrt(m) + A^m/(1 + x*(A +x*O(x^n))^sqrt(m) )^sqrt(m) ) )); round(n!*polcoeff(A, n))}
for(n=0, 30, print1(a(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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