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A291613
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G.f. satisfies: A(x - A(x) + A(x)^2) = x^3.
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1
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1, 1, 1, 3, 8, 24, 70, 213, 661, 2096, 6744, 21979, 72372, 240466, 805176, 2714323, 9204564, 31377860, 107466778, 369613444, 1276043914, 4420532404, 15361787448, 53536660741, 187068856364, 655243469070, 2300251841691, 8091866659762, 28520644071194, 100704832499918, 356180684577830, 1261749324207915, 4476270738313695, 15902368898751100
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OFFSET
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1,4
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) A(x - A(x) + A(x)^2) = x^3.
(2) x - A(x) + A(x)^2 = Ai(x^3), where Ai(A(x)) = x.
a(n) ~ c * d^n / n^(3/2), where d = 3.7158450085937934172961140203365280035... and c = 0.13015040544214577418424... - Vaclav Kotesovec, Aug 28 2017
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EXAMPLE
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G.f.: A(x) = x + x^2 + x^3 + 3*x^4 + 8*x^5 + 24*x^6 + 70*x^7 + 213*x^8 + 661*x^9 + 2096*x^10 + 6744*x^11 + 21979*x^12 + 72372*x^13 + 240466*x^14 + 805176*x^15 + 2714323*x^16 + 9204564*x^17 + 31377860*x^18 + 107466778*x^19 + 369613444*x^20 +...
where A(x - A(x) + A(x)^2) = x^3.
RELATED SERIES.
x - A(x) + A(x)^2 = x^3 - x^6 + x^9 - 3*x^12 + 6*x^15 - 17*x^18 + 48*x^21 - 138*x^24 + 415*x^27 - 1260*x^30 +...
Let Ai(x) be the series reversion of A(x) so that Ai(A(x)) = x, then
Ai(x) = x - x^2 + x^3 - 3*x^4 + 6*x^5 - 17*x^6 + 48*x^7 - 138*x^8 + 415*x^9 - 1260*x^10 + 3897*x^11 - 12216*x^12 + 38703*x^13 - 123837*x^14 + 399440*x^15 - 1297560*x^16 + 4241271*x^17 - 13938900*x^18 + 46033026*x^19 - 152684793*x^20 +...
which satisfies: Ai( Ai(x)^3 ) = Ai(x) - x + x^2.
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PROG
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(PARI) {a(n) = my(A=x, V=[1, 1, 1, 3]); for(i=1, n, V=concat(V, 0); A=x*Ser(V); V[#V]=Vec(subst(A, x, x - A + A^2))[#V-2]); V[n]}
for(n=1, 40, 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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