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A004982 a(n) = (2^n/n!) * Product_{k=0..n-1} (4*k + 3).

%I A004982 #61 Dec 28 2024 03:09:28
%S A004982 1,6,42,308,2310,17556,134596,1038312,8046918,62587140,488179692,
%T A004982 3816677592,29897307804,234578876616,1843119744840,14499208659408,
%U A004982 114181268192838,900017055167076,7100134546318044,56053693786721400,442824180915099060,3500419715805068760,27685137752276452920
%N A004982 a(n) = (2^n/n!) * Product_{k=0..n-1} (4*k + 3).
%C A004982 Conjecture: a(p*n) = a(n) (mod p^2) for prime p == 1 (mod 4) and all positive integers n. Cf. A004981. - _Peter Bala_, Dec 22 2019
%H A004982 Alois P. Heinz, <a href="/A004982/b004982.txt">Table of n, a(n) for n = 0..400</a>
%F A004982 G.f.: (1 - 8*x)^(-3/4).
%F A004982 a(n) ~ Gamma(3/4)^-1*n^(-1/4)*2^(3*n)*{1 - 3/32*n^-1 + ...}
%F A004982 a(n) = 8^n*Gamma(n+3/4)/(n!*Gamma(3/4)). - _Vaclav Kotesovec_, Sep 15 2013
%F A004982 From _Karol A. Penson_, Dec 19 2015: (Start)
%F A004982 a(n) = (-8)^n*binomial(-3/4,n).
%F A004982 E.g.f.: is the hypergeometric function of type 1F1, in Maple notation hypergeom([3/4], [1], 8*x).
%F A004982 Representation as n-th moment of a positive function on (0, 8): a(n) = Integral_{x=0..8} ( x^n*2^(1/4)/(8*Pi*x^(1/4)*(1-x/8)^(3/4)) ) dx, n >= 0. 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)
%F A004982 D-finite with recurrence: n*a(n) +2*(-4*n+1)*a(n-1)=0. - _R. J. Mathar_, Jan 16 2020
%p A004982 A004982 := n -> (-8)^n*binomial(-3/4, n):
%p A004982 seq(A004982(n), n=0..25); # _Peter Luschny_, Oct 23 2018
%t A004982 Table[2^n/n! Product[4k+3,{k,0,n-1}],{n,0,30}] (* _Harvey P. Dale_, Oct 03 2011 *)
%t A004982 Table[Sum[2^k*Binomial[2*n-2*k,n-k]*Binomial[n+k,n],{k,0,n}],{n,0,25}] (* _Vaclav Kotesovec_, Sep 15 2013 *)
%t A004982 FullSimplify[Table[8^n*Gamma[n+3/4]/(n!*Gamma[3/4]), {n, 0, 25}]] (* _Vaclav Kotesovec_, Sep 15 2013 *)
%t A004982 max = 30; s = Hypergeometric1F1[3/4, 1, 8x] + O[x]^(max+1);
%t A004982 CoefficientList[s, x]*(Range[0, max]!) (* _Jean-François Alcover_, Dec 19 2015, after _Karol A. Penson_ *)
%o A004982 (PARI) a(n)=2^n/n!*prod(k=0,n-1,4*k+3)
%o A004982 for(n=0,25,print(a(n)))
%o A004982 (PARI) x='x+O('x^66); Vec((1-8*x)^(-3/4)) \\ _Joerg Arndt_, Apr 20 2013
%o A004982 (Magma) [1] cat [2^n*&*[4*k+3: k in [0..n-1]]/Factorial(n): n in [1..25]]; // _G. C. Greubel_, Aug 22 2019
%o A004982 (Sage) [8^n*rising_factorial(3/4, n)/factorial(n) for n in (0..25)] # _G. C. Greubel_, Aug 22 2019
%o A004982 (GAP) List([0..25], n-> 2^n*Product([0..n-1], k-> 4*k+3)/Factorial(n) ); # _G. C. Greubel_, Aug 22 2019
%Y A004982 Main diagonal of A067001. Cf. A004981.
%K A004982 nonn,easy
%O A004982 0,2
%A A004982 Joe Keane (jgk(AT)jgk.org)
%E A004982 More terms from _Rick L. Shepherd_, Mar 03 2002