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A135149 A binomial recursion: a(n) = p(n) (see formula).

%I A135149 #23 Jan 29 2025 07:55:59
%S A135149 1,5,36,304,2973,33156,415962,5803307,89172846,1496858836,27258427263,
%T A135149 535299208890,11277600621714,253741796354921,6072776118043704,
%U A135149 154050364873902628,4128986249628307077,116598919802471049936,3460199566405679555310,107659401911343963741971
%N A135149 A binomial recursion: a(n) = p(n) (see formula).
%D A135149 Benoit Cloitre, Binomial recursions, Pi and log2, in preparation 2007.
%H A135149 Vaclav Kotesovec, <a href="/A135149/b135149.txt">Table of n, a(n) for n = 1..400</a>
%F A135149 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 A135149 Limit_{n->oo} p(n)/q(n) = (15*Pi - 22)/(52 - 15*Pi) = 5.1524450418835554775446337...
%F A135149 a(n) ~ 2 * (15*Pi - 22) * n^(3/2) * n! / (225 * sqrt(Pi) * log(2)^(n + 5/2)). - _Vaclav Kotesovec_, Nov 25 2020
%F A135149 E.g.f.: exp(5*x/2) * (60*arcsin(exp(x/2)/sqrt(2)) - 22 - 15*Pi) / (150*(2 - exp(x))^(5/2)) + (24*(-3 + 5*x) - 8*exp(x)*(-4 + 15*x) + 2*exp(2*x)*(31 + 15*x)) / (150*(2 - exp(x))^2). - _Vaclav Kotesovec_, Nov 25 2020
%t A135149 z[1] := x; z[n_] := 1 + Sum[(3 + Binomial[n, k])*z[k], {k, 1, n - 1}]; Table[ Coefficient[z[n], x, 1], {n, 1, 10}] (* _G. C. Greubel_, Sep 28 2016 *)
%t A135149 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], {n, 1, 30}] (* _Vaclav Kotesovec_, Nov 25 2020 *)
%t A135149 nmax = 30; Rest[Simplify[CoefficientList[Series[E^(5*x/2)*(60*ArcSin[E^(x/2) / Sqrt[2]] - 22 - 15*Pi) / (150*(2 - E^x)^(5/2)) + (24*(-3 + 5*x) - 8*E^x*(-4 + 15*x) + 2*E^(2*x)*(31 + 15*x))/(150*(2 - E^x)^2), {x, 0, nmax}], x] * Range[0, nmax]!]] (* _Vaclav Kotesovec_, Nov 25 2020 *)
%o A135149 (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 A135149 Cf. A135147, A135148, A135150, A135074, A135075.
%K A135149 nonn
%O A135149 1,2
%A A135149 _Benoit Cloitre_, Nov 20 2007
%E A135149 More terms from _Vaclav Kotesovec_, Nov 25 2020