A024216 a(n) = n-th elementary symmetric function of the first n+1 positive integers congruent to 1 mod 3.
1, 5, 39, 418, 5714, 95064, 1864456, 42124592, 1077459120, 30777463360, 971142388160, 33547112941440, 1259204418129280, 51032742579123200, 2220990565060377600, 103308619261574809600, 5114702794181847910400
Offset: 0
Examples
From _Gheorghe Coserea_, Dec 24 2015: (Start) For n = 1 we have a(1) = 1*4*(1/1 + 1/4) = 5. For n = 2 we have a(2) = 1*4*7*(1/1 + 1/4 + 1/7) = 39. For n = 3 we have a(3) = 1*4*7*10*(1/1 + 1/4 + 1/7 + 1/10) = 418. (End)
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Programs
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Magma
I:=[5,39]; [1] cat [n le 2 select I[n] else (6*n-1) * Self(n-1) - (3*n-2)^2 * Self(n-2) : n in [1..30]]; // Vincenzo Librandi, Aug 30 2015
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Maple
f:= gfun:-rectoproc({-(3*n+1)^2*a(n-1)+(6*n+5)*a(n)-a(n+1), a(0) = 1, a(1) = 5, a(2) = 39}, a(n), remember): map(f, [$0..30]); # Robert Israel, Aug 30 2015 -
Mathematica
Rest[CoefficientList[Series[-(1/3)*Log[1-3*x]/(1-3*x)^(1/3), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Oct 07 2013 *) -
PARI
n = 33; a = vector(n); a[1] = 5; a[2] = 39; for (k = 2, n-1, a[k+1] = (6*k+5) * a[k] - (3*k+1)^2 * a[k-1]); print(concat(1,a)); \\ Gheorghe Coserea, Aug 29 2015
Formula
E.g.f. (for offset 1): -(1/3)*log(1-3*x)/(1-3*x)^(1/3). - Vladeta Jovovic, Sep 26 2003
For n >= 1, a(n-1) = 3^(n-1)*n!*Sum_{k=0..n-1} binomial(k-2/3, k)/(n-k). - Milan Janjic, Dec 14 2008, corrected by Peter Bala, Oct 08 2013
a(n) ~ (n+1)! * GAMMA(2/3) * 3^(n+3/2) * (log(n) + gamma + Pi*sqrt(3)/6 + 3*log(3)/2) / (6*Pi*n^(2/3)), where "GAMMA" is the Gamma function and "gamma" is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 07 2013
a(n+1) = (6*n+5) * a(n) - (3*n+1)^2 * a(n-1). - Gheorghe Coserea, Aug 29 2015
E.g.f.: (3 - log(1-3*x))/(3*(1-3*x)^(4/3)). - Robert Israel, Aug 30 2015
a(n) = A286718(n+1, 1), n >= 0.
Boas-Buck type recurrence: a(0) = 1 and for n >= 1: a(n) = ((n+1)!/n) * Sum_{p=1..n} 3^(n-p)*(1 + 3*beta(n-p))*a(p-1)/p!, with beta(k) = A002208(k+1) / A002209(k+1). Proof from a(n) = A286718(n+1, 1). - Wolfdieter Lang, Aug 09 2017
Extensions
More terms from Vladeta Jovovic, Sep 26 2003
Comments