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A325288
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E.g.f. A(x) satisfies exp( A(x) ) = Sum_{n>=0} exp(n^2*x*A(x))/n!.
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0
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1, 2, 19, 375, 11597, 494073, 26973213, 1802125335, 142754111145, 13101722399345, 1369021004761137, 160654614141341015, 20941910928902384125, 3005307338361684404233, 471283774693207248796085, 80257308377333916859891447, 14763641606278195049365601105, 2920389472814753357684396569825, 618775227663522741562543469882841, 139960830633528124453000154719908471, 33697149359136769906189799588782996325
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OFFSET
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0,2
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COMMENTS
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For comparison, we have exp(W(x)) = Sum_{n>=0} exp(n*x*W(x))/n! where W(x) = exp(x*W(x)) = LambertW(-x)/(-x).
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LINKS
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EXAMPLE
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E.g.f.: A(x) = 1 + 2*x + 19*x^2/2! + 375*x^3/3! + 11597*x^4/4! + 494073*x^5/5! + 26973213*x^6/6! + 1802125335*x^7/7! + 142754111145*x^8/8! + 13101722399345*x^9/9! + 1369021004761137*x^10/10! + 160654614141341015*x^11/11! + 20941910928902384125*x^12/12! + 3005307338361684404233*x^13/13! + 471283774693207248796085*x^14/14! + 80257308377333916859891447*x^15/15! + ...
satisfies
exp(A(x)) = 1 + exp(x*A(x)) + exp(2^2*x*A(x))/2! + exp(3^2*x*A(x))/3! + exp(4^2*x*A(x))/4! + exp(5^2*x*A(x))/5! + exp(6^2*x*A(x))/6! + exp(7^2*x*A(x))/7! + exp(8^2*x*A(x))/8! + ...
RELATED SERIES.
exp(A(x) - 1) = 1 + 2*x + 23*x^2/2! + 497*x^3/3! + 16152*x^4/4! + 708675*x^5/5! + 39396793*x^6/6! + 2661970792*x^7/7! + 212306874267*x^8/8! + 19558829912359*x^9/9! + 2047036462062752*x^10/10! + ...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = round( (#A-1)!*1000*polcoeff( ( exp(-1)*sum(n=0, 20*#A+500, exp(n^2*x*truncate(Ser(A)) +x*O(x^#A))/n!*1.) - exp(Ser(A)-1) ), #A-1) )/(#A-1)!/1000); n!*A[n+1]}
for(n=0, 20, 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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