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A004982
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a(n) = (2^n/n!) * Product_{k=0..n-1} (4*k + 3).
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9
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1, 6, 42, 308, 2310, 17556, 134596, 1038312, 8046918, 62587140, 488179692, 3816677592, 29897307804, 234578876616, 1843119744840, 14499208659408, 114181268192838, 900017055167076, 7100134546318044, 56053693786721400, 442824180915099060, 3500419715805068760, 27685137752276452920
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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FORMULA
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G.f.: (1 - 8*x)^(-3/4).
a(n) ~ Gamma(3/4)^-1*n^(-1/4)*2^(3*n)*{1 - 3/32*n^-1 + ...}
a(n) = (-8)^n*binomial(-3/4,n).
E.g.f.: is the hypergeometric function of type 1F1, in Maple notation hypergeom([3/4], [1], 8*x).
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)
D-finite with recurrence: n*a(n) +2*(-4*n+1)*a(n-1)=0. - R. J. Mathar, Jan 16 2020
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MAPLE
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A004982 := n -> (-8)^n*binomial(-3/4, n):
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MATHEMATICA
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Table[2^n/n! Product[4k+3, {k, 0, n-1}], {n, 0, 30}] (* Harvey P. Dale, Oct 03 2011 *)
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 *)
FullSimplify[Table[8^n*Gamma[n+3/4]/(n!*Gamma[3/4]), {n, 0, 25}]] (* Vaclav Kotesovec, Sep 15 2013 *)
max = 30; s = Hypergeometric1F1[3/4, 1, 8x] + O[x]^(max+1);
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PROG
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(PARI) a(n)=2^n/n!*prod(k=0, n-1, 4*k+3)
for(n=0, 25, print(a(n)))
(PARI) x='x+O('x^66); Vec((1-8*x)^(-3/4)) \\ Joerg Arndt, Apr 20 2013
(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
(Sage) [8^n*rising_factorial(3/4, 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+3)/Factorial(n) ); # G. C. Greubel, Aug 22 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Joe Keane (jgk(AT)jgk.org)
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EXTENSIONS
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STATUS
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approved
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