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 A218221 G.f.: 1/(1-x/(1-4*x/(1-10*x/(1-20*x/(1-35*x/(1-56*x/(1-...))))))), a continued fraction. 1
 1, 1, 5, 65, 1725, 81225, 6181125, 710984625, 117537778125, 26848583825625, 8210318193703125, 3275053250628290625, 1667519951972905828125, 1063947235962694359515625, 837322677987349287566953125, 801714108831393845941434140625 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..198 FORMULA a(n) == 0 (mod 5) for n>1. a(n) == 0 (mod 3) for n>3. G.f.: Q(0), where Q(k) = 1 - x*(k+1)*(k+2)*(k+3)/(x*(k+1)*(k+2)*(k+3) - 6/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 19 2013 a(n) ~ c * d^n * (n!)^3 * n^(5/2), where d = 32*sqrt(3)*Pi^3 / Gamma(1/3)^9 = 0.241821816937867322064737317131437... and c = 2^(15/2) * 3^(11/4) * Pi^4 / Gamma(1/3)^(27/2) = 0.60382458700655692263505976... - Vaclav Kotesovec, Aug 25 2017, updated Mar 16 2024 EXAMPLE G.f.: A(x) = 1 + x + 5*x^2 + 65*x^3 + 1725*x^4 + 81225*x^5 +... MATHEMATICA nmax = 20; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[Binomial[Range[nmax + 1]+2, 3]*x]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2017 *) eulerCF[f_, len_] := Module[{g}, g[len - 1] = 1; g[k_] := g[k] = 1 - f[k]/(f[k] - 2/g[k + 1]); CoefficientList[g[0] + O[x]^len, x]]; A218221List[len_] := eulerCF[(1/3) x (# + 1) (# + 2) (# + 3) &, len]; A218221List[16] (* Peter Luschny, Aug 09 2019 *) PROG (PARI) {a(n)=local(CF=1+x*O(x^n)); for(k=1, n, CF=1/(1-(n-k+1)*(n-k+2)*(n-k+3)/6*x*CF)); polcoeff(CF, n)} for(n=0, 20, print1(a(n), ", ")) CROSSREFS Cf. A000292, A002105, A216966. Sequence in context: A277347 A276755 A346115 * A046881 A336674 A300489 Adjacent sequences: A218218 A218219 A218220 * A218222 A218223 A218224 KEYWORD nonn AUTHOR Paul D. Hanna, Oct 23 2012 STATUS approved

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Last modified August 14 04:31 EDT 2024. Contains 375146 sequences. (Running on oeis4.)