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

0,2

Comments

Comment by R. J. Mathar, Oct 01 2016: (Start)

The k-th elementary symmetric functions of the integers 1+j*3, j=0..n-1, form a triangle T(n,k), 0 <= k <= n, n >= 0:

1

1 1

1 5 4

1 12 39 28

1 22 159 418 280

1 35 445 2485 5714 3640

1 51 1005 9605 45474 95064 58240

1 70 1974 28700 227969 959070 1864456 1106560

1 92 3514 72128 859369 5974388 22963996 42124592 24344320

This here is the first subdiagonal. The diagonal seems to be A007559. The first columns are A000012, A000326, A024212, A024213, A024214. (End)

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

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

Example

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)

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 *)

Prog

(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
(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

Crossrefs

Cf. A024395, A024382, A286718 (first column).

Sequence in context: A124549 A308939 A317618 * A127189 A121354 A122486

Adjacent sequences: A024213 A024214 A024215 * A024217 A024218 A024219

Keyword

nonn,easy

Author

Clark Kimberling

Extensions

More terms from Vladeta Jovovic, Sep 26 2003

Status

approved