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A305604
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G.f. A(x) satisfies: A(x) = 1 + x*[d/dx 1/(1 - x*A(x)^4)].
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3
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1, 1, 10, 165, 3620, 96600, 2996586, 105222740, 4110953640, 176563668420, 8262487393200, 418357209052255, 22790291724038580, 1329373540730613885, 82689132241752729840, 5464831115387029530825, 382492359049214499350160, 28269306654212844076331775, 2200386909130514596330158960, 179936864840741390285785502825, 15424543660650808111807757322400
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OFFSET
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0,3
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LINKS
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FORMULA
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O.g.f. A(x) satisfies:
(1) [x^n] exp( n * Integral A(x)^4 dx ) * (n + 1 - A(x)) = 0 for n > 0.
(2) A(x) = 1 + x*A(x)^3*(A(x) + 4*x*A'(x))/(1 - x*A(x)^4)^2.
a(n) ~ c * 4^n * n^(5/4) * n!, where c = 0.147639333661398142298711... - Vaclav Kotesovec, Oct 06 2020
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EXAMPLE
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G.f.: A(x) = 1 + x + 10*x^2 + 165*x^3 + 3620*x^4 + 96600*x^5 + 2996586*x^6 + 105222740*x^7 + 4110953640*x^8 + 176563668420*x^9 + ...
such that A(x) = 1 + x*[d/dx 1/(1 - x*A(x)^4)].
RELATED SERIES.
A(x)^4 = 1 + 4*x + 46*x^2 + 784*x^3 + 17181*x^4 + 452860*x^5 + 13831594*x^6 + 478200572*x^7 + 18418253542*x^8 + 781180290204*x^9 + ...
1/(1 - x*A(x)^4) = 1 + x + 5*x^2 + 55*x^3 + 905*x^4 + 19320*x^5 + 499431*x^6 + 15031820*x^7 + 513869205*x^8 + 19618185380*x^9 + ...
A'(x)/A(x) = 1 + 19*x + 466*x^2 + 13659*x^3 + 457926*x^4 + 17142730*x^5 + 706064549*x^6 + 31677960427*x^7 + 1537022113117*x^8 + ...
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*deriv(1/(1 - x*A^4+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)*intformal(Ser(A)^4) ) * ((m-1) + 1 - Ser(A)) )[m] ); A[n+1]}
for(n=0, 25, 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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