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A135149 A binomial recursion : a(n) = p(n) (see formula). 6
1, 5, 36, 304, 2973, 33156, 415962, 5803307, 89172846, 1496858836, 27258427263, 535299208890, 11277600621714, 253741796354921, 6072776118043704, 154050364873902628, 4128986249628307077, 116598919802471049936, 3460199566405679555310, 107659401911343963741971 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

B. Cloitre, Binomial recursions, Pi and log2, in preparation 2007

LINKS

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

FORMULA

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).

Lim n-->infty p(n)/q(n) = (15*Pi - 22)/(52 - 15*Pi) = 5.1524450418835554775446337...

a(n) ~ 2 * (15*Pi - 22) * n^(3/2) * n! / (225 * sqrt(Pi) * log(2)^(n + 5/2)). - Vaclav Kotesovec, Nov 25 2020

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

MATHEMATICA

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 *)

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 *)

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 *)

PROG

(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);

CROSSREFS

Cf. A135147, A135148, A135150, A135074, A135075.

Sequence in context: A027331 A255489 A091161 * A269007 A246510 A341961

Adjacent sequences:  A135146 A135147 A135148 * A135150 A135151 A135152

KEYWORD

nonn

AUTHOR

Benoit Cloitre, Nov 20 2007

EXTENSIONS

More terms from Vaclav Kotesovec, Nov 25 2020

STATUS

approved

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Last modified May 23 08:14 EDT 2022. Contains 353961 sequences. (Running on oeis4.)