A051583 a(n) = (2*n+9)!!/9!!, related to A001147 (odd double factorials).
1, 11, 143, 2145, 36465, 692835, 14549535, 334639305, 8365982625, 225881530875, 6550564395375, 203067496256625, 6701227376468625, 234542958176401875, 8678089452526869375, 338445488648547905625, 13876265034590464130625
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..398
- A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
Crossrefs
Programs
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GAP
List([0..20], n-> Product([0..n-1], j-> 2*j+11) ); # G. C. Greubel, Nov 12 2019
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Magma
[1] cat [(&*[2*j+11: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Nov 12 2019
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Maple
seq(2^n*pochhammer(11/2,n), n = 0..20); # G. C. Greubel, Nov 12 2019
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Mathematica
(2*Range[0,20]+9)!!/945 (* Harvey P. Dale, Apr 10 2019 *) Table[2^n*Pochhammer[11/2, n], {n,0,20}] (* G. C. Greubel, Nov 12 2019 *)
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PARI
vector(20, n, prod(j=0,n-2, 2*j+11) ) \\ G. C. Greubel, Nov 12 2019
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Sage
[product( (2*j+11) for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Nov 12 2019
Formula
a(n) = (2*n+9)!!/9!!.
E.g.f.: 1/(1-2*x)^(11/2).
From Peter Bala, May 26 2017: (Start)
a(n+1) = (2*n + 11)*a(n) with a(0) = 1.
O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 11*x)*A(x) - 1 with A(0) = 1.
G.f. as an S-fraction: A(x) = 1/(1 - 11*x/(1 - 2*x/(1 - 13*x/(1 - 4*x/(1 - 15*x/(1 - 6*x/(1 - ... - (2*n + 9)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 11*x/(1 - 13*x/(1 - 2*x/(1 - 15*x/(1 - 4*x/(1 - 17*x/(1 - 6*x/(1 - ... - (2*n + 11)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 945 * sqrt(e*Pi/2) * erf(1/sqrt(2)) - 1332, where erf is the error function.
Sum_{n>=0} (-1)^n/a(n) = 945 * sqrt(Pi/(2*e)) * erfi(1/sqrt(2)) - 684, where erfi is the imaginary error function. (End)
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