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A007559 Triple factorial numbers (3*n-2)!!! with leading 1 added.
(Formerly M3627)
103
1, 1, 4, 28, 280, 3640, 58240, 1106560, 24344320, 608608000, 17041024000, 528271744000, 17961239296000, 664565853952000, 26582634158080000, 1143053268797440000, 52580450364682240000, 2576442067869429760000, 133974987529210347520000, 7368624314106569113600000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

a(n) is the product of the positive integers k <= 3*n that have k modulo 3 = 1. - Peter Luschny, Jun 23 2011

See A094638 for connections to differential operators. - Tom Copeland, Sep 20 2011

Partial products of A016777. - Reinhard Zumkeller, Sep 20 2013

For n > 2, a(n) is a Zumkeller number. - Ivan N. Ianakiev, Jan 28 2020

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

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=0..100

P. Codara, O. M. D'Antona, P. Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers, arXiv preprint arXiv:1308.1700 [cs.DM], 2013.

S. Goodenough, C. Lavault, On subsets of Riordan subgroups and Heisenberg--Weyl algebra, arXiv preprint arXiv:1404.1894 [cs.DM], 2014.

S. Goodenough, C. Lavault, Overview on Heisenberg—Weyl Algebra and Subsets of Riordan Subgroups, The Electronic Journal of Combinatorics, 22(4) (2015), #P4.16.

Orli Herscovici, Generalized permutations related to the degenerate Eulerian numbers, arXiv preprint arXiv:2007.13205 [math.CO], 2020.

Ivan N. Ianakiev, A simple proof that for n > 2, a(n) is a Zumkeller number

Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.

M. D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications, J. Int. Seq. 13 (2010), 10.6.7, Table 6.3.

FORMULA

a(n) = Product_{k=0..n-1} (3*k + 1).

a(n) = (3*n - 2)!!!, n >= 1, and a(0) = 1.

E.g.f.: (1-3*x)^(-1/3).

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

a(n) = 3^n*Pochhammer(1/3, n).

a(n) = Sum_{k=0..n} (-3)^(n-k)*A048994(n, k). - Philippe Deléham, Oct 29 2005

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

From Gary W. Adamson, Jul 19 2011: (Start)

a(n) = upper left term in M^n, M = a variant of Pascal (1,3) triangle (Cf. A095660); as an infinite square production matrix:

  1, 3, 0, 0, 0,...

  1, 4, 3, 0, 0,...

  1, 5, 7, 3, 0,...

  ...

  a(n+1) = sum of top row terms of M^n. (End)

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

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

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

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

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

O.g.f.: hypergeom([1, 1/3], [], 3*x). - Peter Luschny, Oct 08 2015

a(n) = 3^n * Gamma(n + 1/3)/Gamma(1/3). - Artur Jasinski, Aug 23 2016

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

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

D-finite with recurrence: a(n) +(-3*n+2)*a(n-1)=0, n>=1. - R. J. Mathar, Feb 14 2020

Sum_{n>=1) 1/a(n) = (e/9)^(1/3) * (Gamma(1/3) - Gamma(1/3, 1/3)). - Amiram Eldar, Jun 29 2020

EXAMPLE

G.f. = 1 + x + 4*x^2 + 28*x^3 + 280*x^4 + 3640*x^5 + 58240*x^6 + ...

a(3) = 28 and a(4) = 280; with top row of M^3 = (28, 117, 108, 27), sum = 280.

MAPLE

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

MATHEMATICA

a[ n_] := If[ n < 0, 1 / Product[ k, {k, - 2, 3 n - 1, -3}],

  Product[ k, {k, 1, 3 n - 2, 3}]]; (* Michael Somos, Oct 14 2011 *)

FoldList[Times, 1, Range[1, 100, 3]] (* Harvey P. Dale, Jul 05 2013 *)

Range[0, 19]! CoefficientList[Series[((1 - 3 x)^(-1/3)), {x, 0, 19}], x] (* Vincenzo Librandi, Oct 08 2015 *)

PROG

(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

(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 */

(PARI) x='x+O('x^33); Vec(serlaplace((1-3*x)^(-1/3))) /* Joerg Arndt, Apr 24 2011 */

(Sage)

def A007559(n) : return mul(j for j in range(1, 3*n, 3))

[A007559(n) for n in (0..17)]  # Peter Luschny, May 20 2013

(Haskell)

a007559 n = a007559_list !! n

a007559_list = scanl (*) 1 a016777_list

-- Reinhard Zumkeller, Sep 20 2013

(MAGMA)

b:= func< n | (n lt 2) select n else (3*n-2)*Self(n-1) >;

[1] cat [b(n): n in [1..20]]; // G. C. Greubel, Aug 20 2019

(GAP) List([0..20], n-> Product([0..n-1], k-> 3*k+1 )); # G. C. Greubel, Aug 20 2019

CROSSREFS

Cf. A001147, A004987, A008544, A032031, A051141.

a(n)= A035469(n, 1), n >= 1, (first column of triangle A035469(n, m)).

Cf. A107716. - Gary W. Adamson, Oct 22 2009

Cf. A095660. - Gary W. Adamson, Jul 19 2011

Subsequence of A007661. A007696, A008548.

a(n) = A286718(n,0), n >= 0.

Row sums of A336633.

Sequence in context: A182964 A306228 A178599 * A138208 A071212 A090353

Adjacent sequences:  A007556 A007557 A007558 * A007560 A007561 A007562

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Better description from Wolfdieter Lang

STATUS

approved

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Last modified October 24 10:53 EDT 2020. Contains 337975 sequences. (Running on oeis4.)