OFFSET
0,2
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
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)^(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
EXAMPLE
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)
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Formula (see Mathematica line), correction and more terms from Victor Adamchik (adamchik(AT)cs.cmu.edu), Jul 21 2001
STATUS
approved