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a(n) = n-th elementary symmetric function of the first n+1 positive integers congruent to 2 mod 3.
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%I #39 Jul 09 2019 17:25:43

%S 1,7,66,806,12164,219108,4591600,109795600,2951028000,88084714400,

%T 2891353030400,103521905491200,4015191638617600,167714507921497600,

%U 7506196028811110400,358368551285791692800,18180562447078051328000

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

%C Comment by _R. J. Mathar_, Oct 01 2016 (Start):

%C 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:

%C 1

%C 1 2

%C 1 7 10

%C 1 15 66 80

%C 1 26 231 806 880

%C 1 40 595 4040 12164 12320

%C 1 57 1275 14155 80844 219108 209440

%C 1 77 2415 39655 363944 1835988 4591600 4188800

%C 1 100 4186 95200 1276009 10206700 46819324 109795600 96342400

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

%H Vincenzo Librandi, <a href="/A024395/b024395.txt">Table of n, a(n) for n = 0..200</a>

%F E.g.f. (for offset 1): -(1/3)*log(1-3*x)/(1-3*x)^(2/3). - _Vladeta Jovovic_, Sep 26 2003

%F 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

%F 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

%F a(n+1) = (6*n+7) * a(n) - (3*n+2)^2 * a(n-1). - _Gheorghe Coserea_, Aug 30 2015

%F a(n) = A225470(n+1, 1), n >= 0. - _Wolfdieter Lang_, May 29 2017

%e From _Gheorghe Coserea_, Dec 24 2015: (Start)

%e For n=1 we have a(1) = 2*5*(1/2 + 1/5) = 7.

%e For n=2 we have a(2) = 2*5*8*(1/2 + 1/5 + 1/8) = 66.

%e For n=3 we have a(3) = 2*5*8*11*(1/2 + 1/5 + 1/8 + 1/11) = 806.

%e (End)

%t Table[ (-1)^(n+1)*Sum[(-3)^(n - k) k (-1)^(n - k) StirlingS1[n+1, k + 1], {k, 0, n}], {n, 1, 30}]

%t Join[{1},Table[Module[{c=NestList[3+#&,2,n+1]},Times@@c*Total[1/c]],{n,0,20}]] (* _Harvey P. Dale_, Jul 09 2019 *)

%o (PARI)

%o n = 16; a = vector(n); a[1] = 7; a[2] = 66;

%o for (k=2, n-1, a[k+1] = (6*k+7) * a[k] - (3*k+2)^2 * a[k-1]);

%o print(concat(1,a)) \\ _Gheorghe Coserea_, Aug 30 2015

%Y Cf. A024216, A225470 (second column).

%K nonn,easy

%O 0,2

%A _Clark Kimberling_

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