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%I M3627 #165 Sep 08 2022 08:44:35
%S 1,1,4,28,280,3640,58240,1106560,24344320,608608000,17041024000,
%T 528271744000,17961239296000,664565853952000,26582634158080000,
%U 1143053268797440000,52580450364682240000,2576442067869429760000,133974987529210347520000,7368624314106569113600000
%N Triple factorial numbers (3*n-2)!!! with leading 1 added.
%C a(n) is the number of increasing quaternary trees on n vertices. (See A001147 for ternary and A000142 for binary trees.) - _David Callan_, Mar 30 2007
%C a(n) is the product of the positive integers k <= 3*n that have k modulo 3 = 1. - _Peter Luschny_, Jun 23 2011
%C See A094638 for connections to differential operators. - _Tom Copeland_, Sep 20 2011
%C Partial products of A016777. - _Reinhard Zumkeller_, Sep 20 2013
%C For n > 2, a(n) is a Zumkeller number. - _Ivan N. Ianakiev_, Jan 28 2020
%C a(n) is the number of generalized permutations of length n related to the degenerate Eulerian numbers (see arXiv:2007.13205), cf. A336633. - _Orli Herscovici_, Jul 28 2020
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A007559/b007559.txt">Table of n, a(n) for n=0..100</a>
%H Alexander Burstein and Louis W. Shapiro, <a href="https://arxiv.org/abs/2112.11595">Pseudo-involutions in the Riordan group</a>, arXiv:2112.11595 [math.CO], 2021.
%H P. Codara, O. M. D'Antona and P. Hell, <a href="http://arxiv.org/abs/1308.1700">A simple combinatorial interpretation of certain generalized Bell and Stirling numbers</a>, arXiv preprint arXiv:1308.1700 [cs.DM], 2013.
%H S. Goodenough and C. Lavault, <a href="http://arxiv.org/abs/1404.1894">On subsets of Riordan subgroups and Heisenberg--Weyl algebra</a>, arXiv preprint arXiv:1404.1894 [cs.DM], 2014.
%H S. Goodenough and C. Lavault, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p16/0">Overview on Heisenberg—Weyl Algebra and Subsets of Riordan Subgroups</a>, The Electronic Journal of Combinatorics, 22(4) (2015), #P4.16.
%H Orli Herscovici, <a href="https://arxiv.org/abs/2007.13205"> Generalized permutations related to the degenerate Eulerian numbers</a>, arXiv preprint arXiv:2007.13205 [math.CO], 2020.
%H Ivan N. Ianakiev, <a href="/A007559/a007559.txt">A simple proof that for n > 2, a(n) is a Zumkeller number</a>
%H Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H J.-C. Novelli and J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
%H M. D. Schmidt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Schmidt/multifact.html">Generalized j-Factorial Functions, Polynomials, and Applications</a>, J. Int. Seq. 13 (2010), 10.6.7, Table 6.3.
%F a(n) = Product_{k=0..n-1} (3*k + 1).
%F a(n) = (3*n - 2)!!!, n >= 1, and a(0) = 1.
%F E.g.f.: (1-3*x)^(-1/3).
%F a(n) ~ sqrt(2*Pi)/Gamma(1/3)*n^(-1/6)*(3*n/e)^n*(1 - (1/36)/n - ...). - Joe Keane (jgk(AT)jgk.org), Nov 22 2001
%F a(n) = 3^n*Pochhammer(1/3, n).
%F a(n) = Sum_{k=0..n} (-3)^(n-k)*A048994(n, k). - _Philippe Deléham_, Oct 29 2005
%F a(n) = n! *( Sum_{m=1..n} (m/n)*Sum_{k=1..n-m} (binomial(k, n-m-k) * (-1/3)^(n-m-k)*binomial(k+n-1,n-1) + 1 ), n>1. - _Vladimir Kruchinin_, Aug 09 2010
%F From _Gary W. Adamson_, Jul 19 2011: (Start)
%F a(n) = upper left term in M^n, M = a variant of Pascal (1,3) triangle (Cf. A095660); as an infinite square production matrix:
%F 1, 3, 0, 0, 0,...
%F 1, 4, 3, 0, 0,...
%F 1, 5, 7, 3, 0,...
%F ...
%F a(n+1) = sum of top row terms of M^n. (End)
%F a(n) = (-2)^n*Sum_{k=0..n} (3/2)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - _Mircea Merca_, May 03 2012
%F G.f.: 1/Q(0) where Q(k) = 1 - x*(3*k+1)/( 1 - x*(3*k+3)/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Mar 21 2013
%F G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(3*k+1)/(x*(3*k+1) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 26 2013
%F E.g.f.: E(0)/2, where E(k)= 1 + 1/(1 - x*(3*k+1)/(x*(3*k+1) + (k+1)/E(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 26 2013
%F Let D(x) = 1/sqrt(1 - 2*x) be the e.g.f. for the sequence of double factorial numbers A001147. Then the e.g.f. A(x) for the triple factorial numbers satisfies D( Integral_{t=0..x} A(t) dt ) = A(x). Cf. A007696 and A008548. - _Peter Bala_, Jan 02 2015
%F O.g.f.: hypergeom([1, 1/3], [], 3*x). - _Peter Luschny_, Oct 08 2015
%F a(n) = 3^n * Gamma(n + 1/3)/Gamma(1/3). - _Artur Jasinski_, Aug 23 2016
%F a(n) = sigma[3,1]^{(n)}_n, n >= 0, with the elementary symmetric function of degree n in the n numbers 1, 4, 7, ..., 1+3*(n-1), with sigma[3,1]^{(n)}_0 := 1. See the first formula. - _Wolfdieter Lang_, May 29 2017
%F a(n) = (-1)^n / A008544(n), 0 = a(n)*(+3*a(n+1) -a(n+2)) +a(n+1)*a(n+1) for all n in Z. - _Michael Somos_, Sep 30 2018
%F D-finite with recurrence: a(n) +(-3*n+2)*a(n-1)=0, n>=1. - _R. J. Mathar_, Feb 14 2020
%F Sum_{n>=1) 1/a(n) = (e/9)^(1/3) * (Gamma(1/3) - Gamma(1/3, 1/3)). - _Amiram Eldar_, Jun 29 2020
%e G.f. = 1 + x + 4*x^2 + 28*x^3 + 280*x^4 + 3640*x^5 + 58240*x^6 + ...
%e a(3) = 28 and a(4) = 280; with top row of M^3 = (28, 117, 108, 27), sum = 280.
%p A007559 := n -> mul(k, k = select(k-> k mod 3 = 1, [$1 .. 3*n])): seq(A007559(n), n = 0 .. 17); # _Peter Luschny_, Jun 23 2011
%t a[ n_] := If[ n < 0, 1 / Product[ k, {k, - 2, 3 n - 1, -3}],
%t Product[ k, {k, 1, 3 n - 2, 3}]]; (* _Michael Somos_, Oct 14 2011 *)
%t FoldList[Times,1,Range[1,100,3]] (* _Harvey P. Dale_, Jul 05 2013 *)
%t Range[0, 19]! CoefficientList[Series[((1 - 3 x)^(-1/3)), {x, 0, 19}], x] (* _Vincenzo Librandi_, Oct 08 2015 *)
%o (Maxima) a(n):=if n=1 then 1 else (n)!*(sum(m/n*sum(binomial(k,n-m-k)*(-1/3)^(n-m-k)* binomial (k+n-1,n-1),k,1,n-m),m,1,n)+1); \\ _Vladimir Kruchinin_, Aug 09 2010
%o (PARI) {a(n) = if( n<0, (-1)^n / prod(k=0,-1-n, 3*k + 2), prod(k=0, n-1, 3*k + 1))}; /* _Michael Somos_, Oct 14 2011 */
%o (PARI) x='x+O('x^33); Vec(serlaplace((1-3*x)^(-1/3))) /* _Joerg Arndt_, Apr 24 2011 */
%o (Sage)
%o def A007559(n) : return mul(j for j in range(1,3*n,3))
%o [A007559(n) for n in (0..17)] # _Peter Luschny_, May 20 2013
%o (Haskell)
%o a007559 n = a007559_list !! n
%o a007559_list = scanl (*) 1 a016777_list
%o -- _Reinhard Zumkeller_, Sep 20 2013
%o (Magma)
%o b:= func< n | (n lt 2) select n else (3*n-2)*Self(n-1) >;
%o [1] cat [b(n): n in [1..20]]; // _G. C. Greubel_, Aug 20 2019
%o (GAP) List([0..20], n-> Product([0..n-1], k-> 3*k+1 )); # _G. C. Greubel_, Aug 20 2019
%Y Cf. A001147, A004987, A008544, A032031, A051141.
%Y a(n)= A035469(n, 1), n >= 1, (first column of triangle A035469(n, m)).
%Y Cf. A107716. - _Gary W. Adamson_, Oct 22 2009
%Y Cf. A095660. - _Gary W. Adamson_, Jul 19 2011
%Y Subsequence of A007661. A007696, A008548.
%Y a(n) = A286718(n,0), n >= 0.
%Y Row sums of A336633.
%K nonn,nice,easy
%O 0,3
%A _N. J. A. Sloane_
%E Better description from _Wolfdieter Lang_
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