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A274966
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E.g.f. A(x) satisfies: 1 = ...(((((A(x) - x)^(1/2) - x^2/2!)^(1/3) - x^3/3!)^(1/4) - x^4/4!)^(1/5) - x^5/5!)^(1/6) -...- x^n/n!)^(1/(n+1)) -...
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1
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1, 1, 2, 6, 30, 180, 1380, 11760, 116760, 1288560, 15772680, 211217160, 3070766160, 48126078000, 808938290160, 14511273416640, 276665518649520, 5585442224281920, 119014292440002960, 2668801991050475280, 62817503812807423680, 1548361707766975221120, 39881143737823187479680, 1071331562128332368223360, 29961996486664600243005120, 870964354095824682016202880, 26276077241407778648357894400, 821576644748718055815635297280
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OFFSET
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0,3
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LINKS
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 2*x^2/2! + 6*x^3/3! + 30*x^4/4! + 180*x^5/5! + 1380*x^6/6! + 11760*x^7/7! + 116760*x^8/8! + 1288560*x^9/9! + 15772680*x^10/10! + 211217160*x^11/11! + 3070766160*x^12/12! + 48126078000*x^13/13! + 808938290160*x^14/14! + 14511273416640*x^15/15! +...
Illustration of the definition.
(A(x) - x)^(1/2) = 1 + 1/2*x^2 + 1/2*x^3 + 1/2*x^4 + 1/2*x^5 + 7/12*x^6 + 2/3*x^7 +...
((A(x) - x)^(1/2) - x^2/2!)^(1/3) = 1 + 1/6*x^3 + 1/6*x^4 + 1/6*x^5 + 1/6*x^6 + 1/6*x^7 + 17/96*x^8 + 3/16*x^9 +...
(((A(x) - x)^(1/2) - x^2/2!)^(1/3) - x^3/3!)^(1/4) = 1 + 1/24*x^4 + 1/24*x^5 + 1/24*x^6 + 1/24*x^7 + 1/24*x^8 + 1/24*x^9 + 61/1440*x^10 +...
((((A(x) - x)^(1/2) - x^2/2!)^(1/3) - x^3/3!)^(1/4) - x^4/4!)^(1/5) = 1 + 1/120*x^5 + 1/120*x^6 + 1/120*x^7 + 1/120*x^8 + 1/120*x^9 + 1/120*x^10 + 1/120*x^11 + 289/34560*x^12 +...
Working backwards, we can generate the series A(x) in the following manner.
Start with S = 1 and a fixed integer N>1, and repeat:
for_{n=1..N-1} S = ( S + x^(N-n)/(N-n)! )^(N-n) ;
then series S matches A(x) up to the first N terms.
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PROG
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(PARI) {a(n) = my(A=1 +x*O(x^n)); for(k=0, n, A = A^(n+2-k) + x^(n+1-k)/(n+1-k)!); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* Informal generation of first N>1 terms: */
S(N) = my(A = 1 +O(x^N)); for(n=1, N-1, A = ( A + x^(N-n)/(N-n)! )^(N-n) ); Vec(serlaplace(A)) \\ Paul D. Hanna, Apr 26 2017
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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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