|
| |
|
|
A008545
|
|
Quadruple factorial numbers: Product_{k=0..n-1} (4*k + 3).
|
|
47
|
|
|
|
1, 3, 21, 231, 3465, 65835, 1514205, 40883535, 1267389585, 44358635475, 1729986783525, 74389431691575, 3496303289504025, 178311467764705275, 9807130727058790125, 578620712896468617375, 36453104912477522894625, 2442358029135994033939875
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
a(n-1), n >= 1, enumerates increasing plane (a.k.a. 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 binomial(r + 2, 2) types (like a binomial(r + 2, 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. - Wolfdieter Lang, Oct 05 2007 [corrected by Karen A. Yeats, Jun 17 2013]
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
|
|
|
LINKS
|
|
|
|
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). - Paul Barry, Dec 03 2009
a(n) = (-1)^n*Sum_{k = 0..n} 4^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
D-finite with recurrence: a(n) + (-4*n + 1)*a(n-1) = 0. - R. J. Mathar, Dec 04 2012
G.f.: 1/x - G(0)/(2*x), where G(k)= 1 + 1/(1 - x*(4*k-1)/(x*(4*k-1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 04 2013
a(n) = A225471(n, 0), n >= 0. a(n) = sigma[4,3]^{(n)}_n, with the elementary symmetric function sigma[4,3]^{n}_n of degree n of the n numbers 3, 7, 11, ..., (3 + 4*(n-1)), and sigma[4,3]^{n}_0 := 1. See the formula given in the name. - Wolfdieter Lang, May 29 2017
G.f.: 1/(1 - 3*x - 12*x^2/(1 - 11*x - 56*x^2/(1 - 19*x - 132*x^2/(1 - 27*x - 240*x^2/(1 - ...))))) (Jacobi continued fraction). - Nikolaos Pantelidis, Feb 28 2020
Sum_{n>=0} 1/a(n) = 1 + exp(1/4)*(Gamma(3/4) - Gamma(3/4, 1/4))/sqrt(2). - Amiram Eldar, Dec 18 2022
|
|
|
EXAMPLE
|
G.f. = 1 + 3*x + 21*x^2 + 231*x^3 + 3465*x^4 + 65835*x^5 + 1514205*x^6 + ...
a(3) = sigma[4,3]^{3}_3 = 3*7*11 = 231. See the name. - Wolfdieter Lang, May 29 2017
|
|
|
MAPLE
|
f := n->product( (4*k-1), k=0..n);
|
|
|
MATHEMATICA
|
FoldList[Times, 1, 4 Range[0, 20] + 3] (* Harvey P. Dale, Jan 19 2013 *)
a[n_]:= If[n < 0, 1 / Product[ -k, {k, 1, -4 n - 3, 4}], Product[k, {k, 3, 4 n - 1, 4}]]; (* Michael Somos, Jan 17 2014 *)
|
|
|
PROG
|
(Haskell)
a008545 n = a008545_list !! n
a008545_list = scanl (*) 1 a004767_list
(PARI) {a(n) = if( n<0, 1 / prod(k=1, -n, 3 - 4*k), prod(k=1, n, 4*k - 1))}; /* Michael Somos, Jan 17 2014 */
(Magma) [1] cat [(&*[4*k+3: k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 18 2019
(Sage) [product(4*k+3 for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 18 2019
(GAP) List([0..20], n-> Product([0..n-1], k-> 4*k+3) ); # G. C. Greubel, Aug 18 2019
|
|
|
CROSSREFS
|
a(n)= A000369(n+1, 1) (first column of triangle).
|
|
|
KEYWORD
|
nonn,easy,nice
|
|
|
AUTHOR
|
Joe Keane (jgk(AT)jgk.org)
|
|
|
STATUS
|
approved
|
| |
|
|
