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%I #69 Sep 08 2022 08:44:33
%S 1,2,10,60,390,2652,18564,132600,961350,7049900,52169260,388898120,
%T 2916735900,21987701400,166478310600,1265235160560,9647418099270,
%U 73774373700300,565603531702300,4346216612028200,33465867912617140,258165266754475080,1994913424920943800
%N a(n) = (2^n/n!) * Product_{k=0..n-1} (4*k + 1).
%C The convolution of this sequence with itself yields A059304. - _T. D. Noe_, Jun 11 2002
%C 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
%H Vincenzo Librandi, <a href="/A004981/b004981.txt">Table of n, a(n) for n = 0..1000</a>
%H A. Vogt, <a href="http://arxiv.org/abs/1108.2993">Resummation of small-x double logarithms in QCD: semi-inclusive electron-positron annihilation</a>, arXiv preprint arXiv:1108.2993 [hep-ph], 2011.
%F a(n) ~ Gamma(1/4)^-1*n^(-3/4)*2^(3*n)*{1 - 3/32*n^-1 - ...}
%F G.f.: (1-8*x)^(-1/4).
%F A002897(n) = Sum_{k=0..n} a(k)^2*a(n-k)^2. - _Michael Somos_, Jan 31 2007
%F 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
%F D-finite with recurrence: n*a(n) = 2*(4*n-3)*a(n-1). - _R. J. Mathar_, Mar 14 2014
%F From _Karol A. Penson_, Dec 19 2015: (Start)
%F a(n) = (-8)^n*binomial(-1/4,n).
%F E.g.f.: is the hypergeometric function of type 1F1, in Maple notation hypergeom([1/4], [1], 8*x).
%F 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)
%p A004981 := n -> (-8)^n*binomial(-1/4, n):
%p seq(A004981(n), n=0..25); # _Peter Luschny_, Oct 23 2018
%t CoefficientList[Series[(1-8x)^(-1/4), {x, 0, 25}], x] (* _Vincenzo Librandi_, Mar 16 2014 *)
%t Table[8^n*Pochhammer[1/4, n]/n!, {n,0,25}] (* _G. C. Greubel_, Aug 22 2019 *)
%o (PARI) a(n)=if(n<0,0,prod(k=1,n,(8*k-6)/k))
%o (PARI) {a(n)=if(n<0, 0, polcoeff( (1-8*x+x*O(x^n))^(-1/4), n))} /* _Michael Somos_, Jan 31 2007 */
%o (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 */
%o (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
%o (Sage) [8^n*rising_factorial(1/4, n)/factorial(n) for n in (0..25)] # _G. C. Greubel_, Aug 22 2019
%o (GAP) List([0..25], n-> 2^n*Product([0..n-1], k-> 4*k+1)/Factorial(n) ); # _G. C. Greubel_, Aug 22 2019
%Y Cf. A002897, A004982, A059304, A127776, A298799.
%K nonn,easy
%O 0,2
%A Joe Keane (jgk(AT)jgk.org)
%E More terms from _James A. Sellers_, May 01 2000
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