A007696 Quartic (or 4-fold) factorial numbers: a(n) = Product_{k = 0..n-1} (4*k + 1). (Formerly M4001)

1, 1, 5, 45, 585, 9945, 208845, 5221125, 151412625, 4996616625, 184874815125, 7579867420125, 341094033905625, 16713607661375625, 885821206052908125, 50491808745015763125, 3080000333445961550625, 200200021673987500790625, 13813801495505137554553125

Offset

0,3

Comments

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

Hankel transform is A169619. - Paul Barry, Dec 03 2009

References

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

Links

T. D. Noe, Table of n, a(n) for n = 0..100

Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seq. 3 (2000), Article 00.2.4.

J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions, arXiv:1403.5962 [math.CO], 2014.

Maxie D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications , J. Integer Seq. 13 (2010), Article 10.6.7; see page 39.

Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integer Seq. 9 (2006), Article 06.1.1.

Formula

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

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

a(n) = Sum_{k = 0..n} (-4)^(n-k) * A048994(n, k). - Philippe Deléham, Oct 29 2005

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

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

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

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

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

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

a(-n) = (-1)^n / A008545(n). - Michael Somos, Jan 17 2014

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

O.g.f.: hypergeom([1, 1/4], [], 4*x). - Peter Luschny, Oct 08 2015

a(n) = A264781(4*n+1, n). - Alois P. Heinz, Nov 24 2015

a(n) = 4^n * Gamma(n + 1/4)/Gamma(1/4). - Artur Jasinski, Aug 23 2016

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

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

Example

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

Maple

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
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

Mathematica

a[ n_]:= Pochhammer[ 1/4, n] 4^n; (* Michael Somos, Jan 17 2014 *)
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 *)
Range[0, 19]! CoefficientList[Series[((1-4x)^(-1/4)), {x, 0, 19}], x] (* Vincenzo Librandi, Oct 08 2015 *)

Prog

(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 */
(Maxima) A007696(n):=prod(4*k+1, k, 0, n-1)$
makelist(A007696(n), n, 0, 30); /* Martin Ettl, Nov 05 2012 */
(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
(Sage) [4^n*rising_factorial(1/4, n) for n in (0..20)] # G. C. Greubel, Aug 15 2019
(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

Crossrefs

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

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

Sequence in context: A243678 A097328 A051539 * A090136 A357322 A090356

Adjacent sequences: A007693 A007694 A007695 * A007697 A007698 A007699

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Better description from Wolfdieter Lang, Dec 11 1999

Status

approved