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%I #105 Jan 02 2023 05:07:25
%S 1,3,21,231,3465,65835,1514205,40883535,1267389585,44358635475,
%T 1729986783525,74389431691575,3496303289504025,178311467764705275,
%U 9807130727058790125,578620712896468617375,36453104912477522894625,2442358029135994033939875
%N Quadruple factorial numbers: Product_{k=0..n-1} (4*k + 3).
%C 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]
%C 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
%H T. D. Noe, <a href="/A008545/b008545.txt">Table of n, a(n) for n = 0..100</a>
%H Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
%H Keiichi Shigechi, <a href="https://arxiv.org/abs/2212.14666">On the lattice of weighted partitions</a>, arXiv:2212.14666 [math.CO], 2022. See p. 27.
%F a(n) = 3*A034176(n) = (4*n-1)(!^4), n >= 1, a(0) := 1.
%F E.g.f.: (1-4*x)^(-3/4).
%F 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
%F 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
%F 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
%F D-finite with recurrence: a(n) + (-4*n + 1)*a(n-1) = 0. - _R. J. Mathar_, Dec 04 2012
%F 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
%F a(-n) = (-1)^n / A007696(n). - _Michael Somos_, Jan 17 2014
%F G.f.: 1/(1 - b(1)*x / (1 - b(2)*x / ...)) where b = A014601. - _Michael Somos_, Jan 17 2014
%F a(n) = 4^n * Gamma(n+3/4) / Gamma(3/4). - _Vaclav Kotesovec_, Jan 28 2015
%F 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
%F 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
%F 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
%e G.f. = 1 + 3*x + 21*x^2 + 231*x^3 + 3465*x^4 + 65835*x^5 + 1514205*x^6 + ...
%e a(3) = sigma[4,3]^{3}_3 = 3*7*11 = 231. See the name. - _Wolfdieter Lang_, May 29 2017
%p f := n->product( (4*k-1),k=0..n);
%p 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
%t FoldList[Times, 1, 4 Range[0, 20] + 3] (* _Harvey P. Dale_, Jan 19 2013 *)
%t a[n_]:= Pochhammer[3/4, n] 4^n; (* _Michael Somos_, Jan 17 2014 *)
%t 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 *)
%o (PARI) a(n)=prod(k=0,n-1,4*k+3) \\ _Charles R Greathouse IV_, Jun 23 2011
%o (Haskell)
%o a008545 n = a008545_list !! n
%o a008545_list = scanl (*) 1 a004767_list
%o -- _Reinhard Zumkeller_, Oct 25 2013
%o (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 */
%o (Magma) [1] cat [(&*[4*k+3: k in [0..n-1]]): n in [1..20]]; // _G. C. Greubel_, Aug 18 2019
%o (Sage) [product(4*k+3 for k in (0..n-1)) for n in (0..20)] # _G. C. Greubel_, Aug 18 2019
%o (GAP) List([0..20], n-> Product([0..n-1], k-> 4*k+3) ); # _G. C. Greubel_, Aug 18 2019
%Y Cf. A004982, A001813, A047053, A051142, A254286.
%Y a(n)= A000369(n+1, 1) (first column of triangle).
%Y Partial products of A004767.
%Y Cf. A007696, A014601, A225471 (first column).
%K nonn,easy,nice
%O 0,2
%A Joe Keane (jgk(AT)jgk.org)
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