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A135149
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A binomial recursion: a(n) = p(n) (see formula).
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6
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1, 5, 36, 304, 2973, 33156, 415962, 5803307, 89172846, 1496858836, 27258427263, 535299208890, 11277600621714, 253741796354921, 6072776118043704, 154050364873902628, 4128986249628307077, 116598919802471049936, 3460199566405679555310, 107659401911343963741971
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OFFSET
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1,2
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REFERENCES
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B. Cloitre, Binomial recursions, Pi and log2, in preparation 2007.
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LINKS
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FORMULA
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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).
Limit_{n->oo} 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
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MATHEMATICA
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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 *)
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PROG
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(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);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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