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A007559 Triple factorial numbers (3*n-2)!!! with leading 1 added.
(Formerly M3627)
86
1, 1, 4, 28, 280, 3640, 58240, 1106560, 24344320, 608608000, 17041024000, 528271744000, 17961239296000, 664565853952000, 26582634158080000, 1143053268797440000, 52580450364682240000, 2576442067869429760000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) = number of increasing quaternary trees on n vertices. (See A001147 for ternary and A000142 for binary trees.) - David Callan, Mar 30 2007

Starting (1, 4, 28, 280,...) = INVERT transform of A107716: (1, 3, 21,...). - Gary W. Adamson, Oct 22 2009

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

REFERENCES

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

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, 2013

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

W. 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, 2014

FORMULA

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

a(n) = (3*n - 2)!!!

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/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), 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

a(n) = (3n-5)*a(n-2) + (3n-3)*a(n-1), n>=2. - Ivan N. Ianakiev, Aug 08 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( int {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

EXAMPLE

1 + x + 4*x^2 + 28*x^3 + 280*x^4 + 3640*x^5 + 58240*x^6 + 1106560*x^7 + ...

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, 0, 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, 0, prod(k=0, n-1, 3*k + 1))} /* Michael Somos, Oct 14 2011 */

(PARI) x='x+O('x^33); /* that many terms */

Vec(serlaplace((1-3*x)^(-1/3))) /* show terms */ /* 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

CROSSREFS

Cf. A001147, A004987, A032031, A008544, 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

Cf. Subsequence of A007661. A007696, A008548.

Sequence in context: A032274 A182964 A178599 * A138208 A071212 A090353

Adjacent sequences:  A007556 A007557 A007558 * A007560 A007561 A007562

KEYWORD

nonn,nice,easy,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

Better description from Wolfdieter Lang

STATUS

approved

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Last modified August 30 20:11 EDT 2016. Contains 275970 sequences.