A135150 - OEIS

%I #22 Jan 29 2025 08:10:49

%S 0,1,7,59,577,6435,80731,1126321,17306899,290514275,5290386805,

%T 103892269503,2188786203451,49246871008285,1178620260610039,

%U 29898497436003155,801364442718809233,22629823094599476315,671564575318740405283,20894818098241648524577,680161672262047334987995

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

%D Benoit Cloitre, Binomial recursions, Pi and log2, in preparation 2007.

%H Vaclav Kotesovec, <a href="/A135150/b135150.txt">Table of n, a(n) for n = 1..400</a>

%F Let z(1) = x and z(n) = 1 + Sum_{k=1..n-1} (3 + binomial(n,k))*z(k), then z(n) = p(n)*x + q(n).

%F Limit_{n->oo} p(n)/q(n) = (15*Pi - 22)/(52 - 15*Pi) = 5.1524450418835554775446337...

%F a(n) ~ 2 * (52 - 15*Pi) * n^(3/2) * n! / (225 * sqrt(Pi) * log(2)^(n + 5/2)). - _Vaclav Kotesovec_, Nov 25 2020

%F E.g.f.: ((-24 + 44*exp(x) - 46*exp(2*x))/(2 - exp(x))^2 - 15*x + exp(5*x/2)*(52 + 15*Pi - 60*arcsin(exp(x/2)/sqrt(2))) /(2*(2 - exp(x))^(5/2)))/75. - _Vaclav Kotesovec_, Nov 25 2020

%t z[1] := x; z[n_] := 1 + Sum[(3 + Binomial[n, k])*z[k], {k, 1, n - 1}]; Table[

%t Coefficient[z[n], x, 0], {n, 1, 10}] (* _G. C. Greubel_, Sep 28 2016 *)

%t z[1] := x; z[n_] := z[n] = Expand[1 + Sum[(3 + Binomial[n, k])*z[k], {k, 1, n-1}]]; Table[Coefficient[z[n], x, 0], {n, 1, 30}] (* _Vaclav Kotesovec_, Nov 25 2020 *)

%t nmax = 30; Rest[CoefficientList[Series[((-24 + 44*E^x - 46*E^(2*x))/(2 - E^x)^2 - 15*x + E^(5*x/2)*(52 + 15*Pi - 60*ArcSin[E^(x/2)/Sqrt[2]])/(2*(2 - E^x)^(5/2)))/75, {x, 0, nmax}], x] * Range[0, nmax]!] (* _Vaclav Kotesovec_, Nov 25 2020 *)

%o (PARI) r=1; s=3; 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);

%Y Cf. A135147, A135148, A135149, A135074, A135075.

%K nonn,changed

%O 1,3

%A _Benoit Cloitre_, Nov 20 2007

%E More terms from _Vaclav Kotesovec_, Nov 25 2020