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A135148
A binomial recursion: a(n) = q(n) (see formula).
6
0, 1, 6, 45, 400, 4115, 48146, 631729, 9189972, 146829039, 2556200086, 48167698733, 976792093784, 21211601837803, 491112582793626, 12077021182230057, 314362864408454236, 8635229233659916007, 249631741661080132766, 7575921686807827601701, 240827454421807200901728
OFFSET
1,3
LINKS
FORMULA
Let z(1) = x and z(n) = 1 + Sum_{k=1..n-1} ( (2 + binomial(n,k))*z(k)) ), then z(n) = p(n)*x + q(n).
Limit_{n->oo} p(n)/q(n) = (3 - 2*log(2))/(2*log(2) - 1 ) = 4.177398899124179661610768...
a(n) ~ (2*log(2) - 1) * n * n! / (8 * log(2)^(n+2)). - Vaclav Kotesovec, Nov 25 2020
E.g.f.: (1 - exp(x)) * (exp(x) - 2*x - 1) / (2*(2 - exp(x))^2). - Vaclav Kotesovec, Nov 25 2020
MATHEMATICA
z[1] := x; z[n_] := 1 + Sum[(2 + Binomial[n, k])*z[k], {k, 1, n - 1}]; Table[ Coefficient[z[n], x, 0], {n, 1, 20}] (* G. C. Greubel, Sep 28 2016 *)
z[1] := x; z[n_] := z[n] = Expand[1 + Sum[(2 + Binomial[n, k])*z[k], {k, 1, n-1}]]; Table[Coefficient[z[n], x, 0], {n, 1, 30}] (* Vaclav Kotesovec, Nov 25 2020 *)
nmax = 30; Rest[CoefficientList[Series[(1 - E^x)*(E^x - 2*x - 1)/(2*(2 - E^x)^2), {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Nov 25 2020 *)
PROG
(PARI) r=1; s=2; 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);
CROSSREFS
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Nov 20 2007
STATUS
approved