A135075 A binomial recursion : a(n) = q(n) (see formula).

0, 1, 5, 33, 265, 2505, 27261, 335757, 4617461, 70138689, 1166295457, 21072290241, 411069239997, 8611025176533, 192788027607293, 4594027768539585, 116093660372707273, 3101080076109154137, 87305805274735566669

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