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A007559
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Triple factorial numbers (3*n-2)!!! with leading 1 added.
(Formerly M3627)
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69
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1, 1, 4, 28, 280, 3640, 58240, 1106560, 24344320, 608608000, 17041024000, 528271744000, 17961239296000, 664565853952000, 26582634158080000, 1143053268797440000, 52580450364682240000, 2576442067869429760000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n) = number of increasing quaternary trees on n vertices. (See A001147 for ternary and A000142 for binary trees.) - David Callan (callan(AT)stat.wisc.edu), Mar 30 2007
Starting (1, 4, 28, 280,...) = INVERT transform of A107716: (1, 3, 21,...) [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 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
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REFERENCES
| N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
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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 DELEHAM (kolotoko(AT)wanadoo.fr), 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. [From Kruchinin Vladimir (kru(AT)ie.tusur.ru), 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)
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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.
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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
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MATHEMATICA
| a[ n_] := If[ n < 0, 0, Product[k, {k, 1, 3 n - 2, 3}]] (* Michael Somos, Oct 14 2011 *)
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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); [From Kruchinin Vladimir (kru(AT)ie.tusur.ru), 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 */
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CROSSREFS
| Cf. A001147, A004987, A032031, A008544, A051141. a(n)= A035469(n, 1), n >= 1, (first column of triangle A035469(n, m)).
Cf. A107716 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 22 2009]
Cf. A095660 [From Gary W. Adamson, Jul 19 2011]
Sequence in context: A032274 A182964 A178599 * A138208 A071212 A090353
Adjacent sequences: A007556 A007557 A007558 * A007560 A007561 A007562
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Better description from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de).
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