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 A135075 A binomial recursion : a(n) = q(n) (see formula). 8
 0, 1, 5, 33, 265, 2505, 27261, 335757, 4617461, 70138689, 1166295457, 21072290241, 411069239997, 8611025176533, 192788027607293, 4594027768539585, 116093660372707273, 3101080076109154137, 87305805274735566669 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS Robert Israel, Table of n, a(n) for n = 1..423 FORMULA Let z(1) = x and z(n) = 1 + Sum_{k=1,..,n-1} ( (1 + binomial(n,k))*z(k) ), then z(n) = p(n)*x + q(n). Lim n-->infinity p(n)/q(n) = (3*Pi - 14)/ (8 - 3*Pi) = 3.2111824896280692148... E.g.f.: g(x) = ((-3*x-8)*exp(x)+6*x+4)/(9*exp(x)-18) -exp(3*x/2)*(-4*arctan(exp(x/2)/sqrt(2-exp(x)))+Pi+8/3)/(6*(2-exp(x))^(3/2)) satisfies (exp(x)-2) g'(x) + 3 g(x) + x = 0. - Robert Israel, Mar 06 2017 a(n) ~ (3*Pi - 8) * sqrt(n) * n! / (9 * sqrt(Pi) * log(2)^(n + 3/2)). - Vaclav Kotesovec, Nov 25 2020 MAPLE A[1]:= 0: for n from 2 to 50 do A[n]:= 1 + add((1+binomial(n, k))*A[k], k=1..n-1) od: seq(A[i], i=1..50); # Robert Israel, Mar 06 2017 MATHEMATICA z[1] := x; z[n_] := 1 + Sum[(1 + Binomial[n, k])*z[k], {k, 1, n - 1}]; Table[Coefficient[z[n], x, 0], {n, 1, 10}] (* G. C. Greubel, Sep 22 2016 *) z[1] := x; z[n_] := z[n] = Expand[1 + Sum[(1 + Binomial[n, k])*z[k], {k, 1, n-1}]]; Table[Coefficient[z[n], x, 0], {n, 1, 30}] (* Vaclav Kotesovec, Nov 25 2020 *) PROG (PARI) r=1; s=1; v=vector(120, j, x); for(n=2, 120, g=r+sum(k=1, n-1, (s+binomial(n, k))*v[k]); v[n]=g); z(n)=v[n]; p(n)=polcoeff(z(n), 1); q(n)=polcoeff(z(n), 0); a(n)=q(n); CROSSREFS Cf. A135074. Sequence in context: A111530 A367946 A087633 * A269820 A302075 A215671 Adjacent sequences: A135072 A135073 A135074 * A135076 A135077 A135078 KEYWORD nonn AUTHOR Benoit Cloitre, Nov 17 2007 STATUS approved

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Last modified April 17 10:55 EDT 2024. Contains 371763 sequences. (Running on oeis4.)