%I M4001 #134 Mar 08 2024 11:55:11

%S 1,1,5,45,585,9945,208845,5221125,151412625,4996616625,184874815125,

%T 7579867420125,341094033905625,16713607661375625,885821206052908125,

%U 50491808745015763125,3080000333445961550625,200200021673987500790625,13813801495505137554553125

%N Quartic (or 4-fold) factorial numbers: a(n) = Product_{k = 0..n-1} (4*k + 1).

%C a(n), n >= 1, enumerates increasing quintic (5-ary) trees. See David Callan's comment on A007559 (number of increasing quarterny trees).

%C Hankel transform is A169619. - _Paul Barry_, Dec 03 2009

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A007696/b007696.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 Seq. 3 (2000), Article 00.2.4.

%H J.-C. Novelli and J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions</a>, arXiv:1403.5962 [math.CO], 2014.

%H Maxie D. Schmidt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Schmidt/multifact.html">Generalized j-Factorial Functions, Polynomials, and Applications </a>, J. Integer Seq. 13 (2010), Article 10.6.7; see page 39.

%H Michael Z. Spivey and Laura L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, J. Integer Seq. 9 (2006), Article 06.1.1.

%F E.g.f.: (1 - 4*x)^(-1/4).

%F a(n) ~ 2^(5/2) * Pi^(1/2) * Gamma(1/4)^(-1) * n^(3/4) * 2^(2*n) * e^(-n) * n^n * (1 + 23/96 * n^(-1) - ...). - Joe Keane (jgk(AT)jgk.org), Nov 23 2001

%F a(n) = Sum_{k = 0..n} (-4)^(n-k) * A048994(n, k). - _Philippe Deléham_, Oct 29 2005

%F G.f.: 1/(1 - x/(1 - 4*x/(1 - 5*x/(1 - 8*x/(1 - 9*x/(1 - 12*x/(1 - 13*x/(1 - .../(1 - A042948(n+1)*x/(1 -... (continued fraction). - _Paul Barry_, Dec 03 2009

%F a(n) = (-3)^n * Sum_{k = 0..n} (4/3)^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 G.f.: 1/T(0), where T(k) = 1 - x * (4*k + 1)/(1 - x * (4*k + 4)/T(k+1)) (continued fraction). - _Sergei N. Gladkovskii_, Mar 19 2013

%F G.f.: 1 + x/Q(0), where Q(k) = 1 + x + 2*(2*k - 1)*x - 4*x*(k+1)/Q(k+1) (continued fraction). - _Sergei N. Gladkovskii_, May 03 2013

%F G.f.: G(0)/2, 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 0 = a(n) * (4*a(n+1) - a(n+2)) + a(n+1) * a(n+1) for all n in Z. - _Michael Somos_, Jan 17 2014

%F a(-n) = (-1)^n / A008545(n). - _Michael Somos_, Jan 17 2014

%F Let T(x) = 1/(1 - 3*x)^(1/3) be the e.g.f. for the sequence of triple factorial numbers A007559. Then the e.g.f. A(x) for the quartic factorial numbers satisfies T(int_{0..x} A(t) dt) = A(x). (Cf. A007559 and A008548.) - _Peter Bala_, Jan 02 2015

%F O.g.f.: hypergeom([1, 1/4], [], 4*x). - _Peter Luschny_, Oct 08 2015

%F a(n) = A264781(4*n+1, n). - _Alois P. Heinz_, Nov 24 2015

%F a(n) = 4^n * Gamma(n + 1/4)/Gamma(1/4). - _Artur Jasinski_, Aug 23 2016

%F D-finite with recurrence: a(n) +(-4*n+3)*a(n-1)=0, n>=1. - _R. J. Mathar_, Feb 14 2020

%F Sum_{n>=0} 1/a(n) = 1 + exp(1/4)*(Gamma(1/4) - Gamma(1/4, 1/4))/(2*sqrt(2)). - _Amiram Eldar_, Dec 18 2022

%e G.f. = 1 + x + 5*x^2 + 45*x^3 + 585*x^4 + 9945*x^5 + 208845*x^6 + ...

%p x:='x'; G(x):=(1-4*x)^(-1/4): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od: seq(eval(f[n],x=0),n=0..17);# _Zerinvary Lajos_, Apr 03 2009

%p A007696 := n -> mul(k, k = select(k-> k mod 4 = 1, [$ 1 .. 4*n])): seq(A007696(n), n=0..17); # _Peter Luschny_, Jun 23 2011

%t a[ n_]:= Pochhammer[ 1/4, n] 4^n; (* _Michael Somos_, Jan 17 2014 *)

%t a[ n_]:= If[n < 0, 1 / Product[ -k, {k, 3, -4n-1, 4}], Product[ k, {k, 1, 4n-3, 4}]]; (* _Michael Somos_, Jan 17 2014 *)

%t Range[0, 19]! CoefficientList[Series[((1-4x)^(-1/4)), {x, 0, 19}], x] (* _Vincenzo Librandi_, Oct 08 2015 *)

%o (PARI) {a(n) = if( n<0, 1 / prod(k=1, -n, 1 - 4*k), prod(k=1, n, 4*k - 3))}; /* _Michael Somos_, Jan 17 2014 */

%o (Maxima) A007696(n):=prod(4*k+1,k,0,n-1)$

%o makelist(A007696(n),n,0,30); /* _Martin Ettl_, Nov 05 2012 */

%o (Magma) [n le 2 select 1 else (4*(n-1)-7)*(Self(n-1) + 4*Self(n-2)): n in [1..20]]; // _G. C. Greubel_, Aug 15 2019

%o (Sage) [4^n*rising_factorial(1/4, n) for n in (0..20)] # _G. C. Greubel_, Aug 15 2019

%o (GAP) a:=[1,1];; for n in [3..20] do a[n]:=(4*(n-1)-7)*(a[n-1]+4*a[n-2]); od; a; # _G. C. Greubel_, Aug 15 2019

%Y Cf. A001147, A001813, A004981, A007559, A008545, A034255, A047053, A051142, A264781.

%Y a(n) = A049029(n, 1) for n >= 1 (first column of triangle).

%K nonn

%O 0,3

%A _N. J. A. Sloane_

%E Better description from _Wolfdieter Lang_, Dec 11 1999