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A008545
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Quadruple factorial numbers: product[ k=0..n-1 ] (4*k+3).
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23
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1, 3, 21, 231, 3465, 65835, 1514205, 40883535, 1267389585, 44358635475, 1729986783525, 74389431691575, 3496303289504025, 178311467764705275, 9807130727058790125, 578620712896468617375
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(n-1), n>=1, enumerates increasing plane (aka ordered) trees with n-vertices (one of them a root labeled 1) with one version of a vertex with out-degree r=0 (a leaf or a root) and each vertex with out-degree r>=1 comes in r+2 types (like an (r+2)-ary vertex). See the increasing tree comments under A001498. For example, a(1)=3 from the three trees with n=2 vertices (a root (out-degree r=1, label 1) and a leaf (r=0), label 2). There are three such trees because of the three types of out-degree r=1 vertices. W. Lang Oct 05 2007.
a(n) is the product of the positive integers less than or equal to 4n that have modulo 4 = 3. - Peter Luschny, Jun 23 2011
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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)= 3*A034176(n) = (4*n-1)(!^4), n >= 1, a(0) := 1. E.g.f. (1-4*x)^(-3/4).
a(n) ~ 2^(1/2)*pi^(1/2)*Gamma(3/4)^-1*n^(1/4)*2^(2*n)*e^-n*n^n*{1 - 1/96*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 23 2001
G.f.: 1/(1-3x/(1-4x/(1-7x/(1-8x/(1-11x/(1-12x/(1-15x/(1-16x/(1-19x/(1-20x/(1-23x/(1-24x/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Dec 03 2009]
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MAPLE
| f := n->product( (4*k-1), k=0..n);
A008545 := n -> mul(k, k = select(k-> k mod 4 = 3, [$1 .. 4*n])): seq(A008545(n), n=0..15); - Peter Luschny, Jun 23 2011
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MATHEMATICA
| s=1; lst={s}; Do[s+=n*s; AppendTo[lst, s], {n, 2, 5!, 4}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 08 2008]
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PROG
| (PARI) a(n)=prod(k=0, n-1, 4*k+3) \\ Charles R Greathouse IV, Jun 23 2011
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CROSSREFS
| Cf. A004982, A001813, A047053, A051142. a(n)= A000369(n+1, 1) (first column of triangle).
Sequence in context: A074638 A097329 A119097 * A005373 A078586 A179331
Adjacent sequences: A008542 A008543 A008544 * A008546 A008547 A008548
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KEYWORD
| nonn,easy,nice
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AUTHOR
| Joe Keane (jgk(AT)jgk.org)
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