A004981 a(n) = (2^n/n!) * Product_{k=0..n-1} (4*k + 1).

1, 2, 10, 60, 390, 2652, 18564, 132600, 961350, 7049900, 52169260, 388898120, 2916735900, 21987701400, 166478310600, 1265235160560, 9647418099270, 73774373700300, 565603531702300, 4346216612028200, 33465867912617140, 258165266754475080, 1994913424920943800

Offset

0,2

Comments

The convolution of this sequence with itself yields A059304. - T. D. Noe, Jun 11 2002

Conjecture: a(p*n) = a(n) (mod p^2) for prime p = 1 (mod 4) and all positive integers n. Cf. A004982 and A298799. - Peter Bala, Dec 22 2019

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

A. Vogt, Resummation of small-x double logarithms in QCD: semi-inclusive electron-positron annihilation, arXiv preprint arXiv:1108.2993 [hep-ph], 2011.

Formula

a(n) ~ Gamma(1/4)^-1*n^(-3/4)*2^(3*n)*{1 - 3/32*n^-1 - ...}

G.f.: (1-8*x)^(-1/4).

A002897(n) = Sum_{k=0..n} a(k)^2*a(n-k)^2. - Michael Somos, Jan 31 2007

a(n) = (Sum_{m=1..n} m*Sum_{k=m..n} binomial(-m+2*k-1,k-1)*2^(n+m-k)*binomial(2*n-k-1,n-1))/n, n>0, a(0)=1. - Vladimir Kruchinin, Dec 26 2011

D-finite with recurrence: n*a(n) = 2*(4*n-3)*a(n-1). - R. J. Mathar, Mar 14 2014

From Karol A. Penson, Dec 19 2015: (Start)

a(n) = (-8)^n*binomial(-1/4,n).

E.g.f.: is the hypergeometric function of type 1F1, in Maple notation hypergeom([1/4], [1], 8*x).

Representation as n-th moment of a positive function on (0, 8): a(n)=int(x^n*(sqrt(2)/(16*Pi*(x/8)^(3/4)*(1-x/8)^(1/4))), x=0..8), n=0, 1, ... . This function is the solution of the Hausdorff moment problem on (0, 8) with moments equal to a(n). As a consequence this representation is unique. (End)

Maple

A004981 := n -> (-8)^n*binomial(-1/4, n):
seq(A004981(n), n=0..25); # Peter Luschny, Oct 23 2018

Mathematica

CoefficientList[Series[(1-8x)^(-1/4), {x, 0, 25}], x] (* Vincenzo Librandi, Mar 16 2014 *)
Table[8^n*Pochhammer[1/4, n]/n!, {n, 0, 25}] (* G. C. Greubel, Aug 22 2019 *)

Prog

(PARI) a(n)=if(n<0, 0, prod(k=1, n, (8*k-6)/k))
(PARI) {a(n)=if(n<0, 0, polcoeff( (1-8*x+x*O(x^n))^(-1/4), n))} /* Michael Somos, Jan 31 2007 */
(Maxima) a(n):=if n=0 then 1 else (sum(m*sum(binomial(-m+2*k-1, k-1) *2^(n+m-k)*binomial(2*n-k-1, n-1), k, m, n), m, 1, n))/(n); /* Vladimir Kruchinin, Dec 26 2011 */
(Magma) [1] cat [2^n*&*[4*k+1: k in [0..n-1]]/Factorial(n): n in [1..25]]; // G. C. Greubel, Aug 22 2019
(Sage) [8^n*rising_factorial(1/4, n)/factorial(n) for n in (0..25)] # G. C. Greubel, Aug 22 2019
(GAP) List([0..25], n-> 2^n*Product([0..n-1], k-> 4*k+1)/Factorial(n) ); # G. C. Greubel, Aug 22 2019

Crossrefs

Cf. A002897, A004982, A059304, A127776, A298799.

Sequence in context: A173613 A341924 A350665 * A214764 A365554 A137571

Adjacent sequences: A004978 A004979 A004980 * A004982 A004983 A004984

Keyword

nonn,easy

Author

Joe Keane (jgk(AT)jgk.org)

Extensions

More terms from James A. Sellers, May 01 2000

Status

approved