A135074 A binomial recursion: a(n) is the coefficient of x in z(n), where z(1) = x and z(n) = 1 + Sum_{k=1..n-1} (binomial(n,k) + 1)*z(k) for n > 1.

1, 3, 16, 106, 851, 8044, 87540, 1078177, 14827510, 225228130, 3745187549, 67666969438, 1320018345504, 27651573264631, 619077538462468, 14752261527199414, 372797929345665683, 9958134039336196072, 280354873141108774272, 8297089960595144115505, 257514010200875255884522

Offset: 1

Author

Benoit Cloitre, Nov 17 2007

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..400

Crossrefs

Cf. A135075.

Programs

Mathematica

z[1] := x; z[n_] := 1 + Sum[(1 + Binomial[n, k])*z[k], {k, 1, n - 1}];
Table[Coefficient[z[n], x], {n, 1, 15}] (* 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], {n, 1, 30}] (* Vaclav Kotesovec, Nov 25 2020 *)
nmax = 30; Rest[Simplify[CoefficientList[Series[(E^(3*x/2)*(14 + 3*Pi) - 12*E^(3*x/2)*ArcSin[E^(x/2)/Sqrt[2]]) / (18*(2 - E^x)^(3/2)) - (2*(1 - 3*x) + E^x*(5 + 3*x))/(9*(2 - E^x)), {x, 0, nmax}], x] * Range[0, nmax]!]] (* Vaclav Kotesovec, Nov 25 2020 *)

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)=p(n);

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...

a(n) ~ (14 - 3*Pi) * sqrt(n) * n! / (9 * sqrt(Pi) * log(2)^(n + 3/2)). - Vaclav Kotesovec, Nov 25 2020

E.g.f.: (exp(3*x/2)*(14 + 3*Pi) - 12*exp(3*x/2)*arcsin(exp(x/2)/sqrt(2))) / (18*(2 - exp(x))^(3/2)) - (2*(1 - 3*x) + exp(x)*(5 + 3*x))/(9*(2 - exp(x))). - Vaclav Kotesovec, Nov 25 2020

Extensions

New name from Charles R Greathouse IV, Sep 22 2016

More terms from Amiram Eldar, Nov 25 2020