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 A024395 a(n) = n-th elementary symmetric function of the first n+1 positive integers congruent to 2 mod 3. 5
 1, 7, 66, 806, 12164, 219108, 4591600, 109795600, 2951028000, 88084714400, 2891353030400, 103521905491200, 4015191638617600, 167714507921497600, 7506196028811110400, 358368551285791692800, 18180562447078051328000 (list; graph; refs; listen; history; text; internal format)
 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 This here is the first subdiagonal. The diagonal seems to be A008544. The first columns are A000012, A005449, A024391, A024392. (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)^(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 Cf. A024216, A225470 (second column). Sequence in context: A300991 A122705 A185181 * A215077 A003286 A244602 Adjacent sequences:  A024392 A024393 A024394 * A024396 A024397 A024398 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

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Last modified May 26 20:43 EDT 2022. Contains 354092 sequences. (Running on oeis4.)