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A007559 Triple factorial numbers (3*n-2)!!! with leading 1 added.
(Formerly M3627)
76
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

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) ~ 2^(1/2)*pi^(1/2)*Gamma(1/3)^-1*n^(-1/6)*3^n*e^-n*n^n*{1 - 1/36*n^-1 - ...}. - 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]

Contribution 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

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 *)

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.

Sequence in context: A032274 A182964 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 August 30 04:09 EDT 2014. Contains 246216 sequences.