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A051578
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a(n) = (2*n+4)!!/4!!, related to A000165 (even double factorials).
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13
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1, 6, 48, 480, 5760, 80640, 1290240, 23224320, 464486400, 10218700800, 245248819200, 6376469299200, 178541140377600, 5356234211328000, 171399494762496000, 5827582821924864000, 209792981589295104000, 7972133300393213952000, 318885332015728558080000
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OFFSET
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0,2
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COMMENTS
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Row m=4 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.
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LINKS
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FORMULA
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a(n) = (2*n+4)!!/4!!.
E.g.f.: 1/(1-2*x)^3.
a(n) ~ 2^(-1/2)*Pi^(1/2)*n^(5/2)*2^n*e^-n*n^n*{1 + 37/12*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 23 2001
a(n+1) = (2*n + 6)*a(n) with a(0) = 1.
O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 6*x)*A(x) - 1 with A(0) = 1.
G.f. as an S-fraction: A(x) = 1/(1 - 6*x/(1 - 2*x/(1 - 8*x/(1 - 4*x/(1 - 10*x/(1 - 6*x/(1 - ... - (2*n + 4)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 6*x/(1 - 8*x/(1 - 2*x/(1 - 10*x/(1 - 4*x/(1 - 12*x/(1 - 6*x/(1 - ... - (2*n + 6)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
Sum_{n>=0} 1/a(n) = 8*sqrt(e) - 12.
Sum_{n>=0} (-1)^n/a(n) = 8/sqrt(e) - 4. (End)
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MAPLE
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a:= proc(n) option remember; `if`(n=0, 1, 2*(n+2)*a(n-1)) end:
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MATHEMATICA
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PROG
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(PARI) vector(21, n, 2^(n-2)*(n+1)! ) \\ G. C. Greubel, Nov 11 2019
(Magma) [2^(n-1)*Factorial(n+2): n in [0..20]]; // G. C. Greubel, Nov 11 2019
(Sage) [2^(n-1)*factorial(n+2) for n in (0..20)] # G. C. Greubel, Nov 11 2019
(GAP) List([0..20], n-> 2^(n-1)*Factorial(n+2) ); # G. C. Greubel, Nov 11 2019
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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