A024395 a(n) = n-th elementary symmetric function of the first n+1 positive integers congruent to 2 mod 3.

1, 7, 66, 806, 12164, 219108, 4591600, 109795600, 2951028000, 88084714400, 2891353030400, 103521905491200, 4015191638617600, 167714507921497600, 7506196028811110400, 358368551285791692800, 18180562447078051328000

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

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

The k-th elementary symmetric functions of the integers 2+j*3, j=0..n-1, form a triangle T(n,k), 0<=k<=n, n>=0:

1

1 2

1 7 10

1 15 66 80

1 26 231 806 880

1 40 595 4040 12164 12320

1 57 1275 14155 80844 219108 209440

1 77 2415 39655 363944 1835988 4591600 4188800

1 100 4186 95200 1276009 10206700 46819324 109795600 96342400

This here is the first subdiagonal. The diagonal seems to be A008544. The first columns are A000012, A005449, A024391, A024392. (End)

Examples

			From _Gheorghe Coserea_, Dec 24 2015: (Start)
For n=1 we have a(1) = 2*5*(1/2 + 1/5) = 7.
For n=2 we have a(2) = 2*5*8*(1/2 + 1/5 + 1/8) = 66.
For n=3 we have a(3) = 2*5*8*11*(1/2 + 1/5 + 1/8 + 1/11) = 806.
(End)
		

Links

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

Crossrefs

Cf. A024216, A225470 (second column).

Programs

Mathematica

Table[ (-1)^(n+1)*Sum[(-3)^(n - k) k (-1)^(n - k) StirlingS1[n+1, k + 1], {k, 0, n}], {n, 1, 30}]
Join[{1},Table[Module[{c=NestList[3+#&,2,n+1]},Times@@c*Total[1/c]],{n,0,20}]] (* Harvey P. Dale, Jul 09 2019 *)

PARI

n = 16; a = vector(n); a[1] = 7; a[2] = 66;
for (k=2, n-1, a[k+1] = (6*k+7) * a[k] - (3*k+2)^2 * a[k-1]);
print(concat(1,a))  \\ Gheorghe Coserea, Aug 30 2015

Formula

E.g.f. (for offset 1): -(1/3)*log(1-3*x)/(1-3*x)^(2/3). - Vladeta Jovovic, Sep 26 2003

For n >= 1, a(n-1) = 3^(n-1)*n!*sum(binomial(k-1/3,k)/(n-k), k = 0..n-1). - Milan Janjic, Dec 14 2008, corrected by Peter Bala, Oct 08 2013

a(n) ~ (n+1)! * 3^n * (log(n) + gamma - Pi*sqrt(3)/6 + 3*log(3)/2) / (n^(1/3)*GAMMA(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+7) * a(n) - (3*n+2)^2 * a(n-1). - Gheorghe Coserea, Aug 30 2015

a(n) = A225470(n+1, 1), n >= 0. - Wolfdieter Lang, May 29 2017

Extensions

Formula (see Mathematica line), correction and more terms from Victor Adamchik (adamchik(AT)cs.cmu.edu), Jul 21 2001