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A051581
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a(n) = (2*n+7)!!/7!!, related to A001147 (odd double factorials).
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7
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1, 9, 99, 1287, 19305, 328185, 6235515, 130945815, 3011753745, 75293843625, 2032933777875, 58955079558375, 1827607466309625, 60311046388217625, 2110886623587616875, 78102805072741824375, 3046009397836931150625
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OFFSET
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0,2
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COMMENTS
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Row m=7 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..400
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
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FORMULA
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a(n) = (2*n+7)!!/7!!.
E.g.f.: 1/(1-2*x)^(9/2).
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x/(x + 1/(2*k+9)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 02 2013
From Peter Bala, May 26 2017: (Start)
a(n+1) = (2*n + 9)*a(n) with a(0) = 1.
O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 9*x)*A(x) - 1 with A(0) = 1.
G.f. as an S-fraction: A(x) = 1/(1 - 9*x/(1 - 2*x/(1 - 11*x/(1 - 4*x/(1 - 13*x/(1 - 6*x/(1 - ... - (2*n + 7)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 9*x/(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 - ...)))))))))). (End)
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MAPLE
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df:=doublefactorial; seq(df(2*n+7)/df(7), n = 0..20); # G. C. Greubel, Nov 12 2019
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MATHEMATICA
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Table[2^n*Pochhammer[9/2, n], {n, 0, 20}] (* G. C. Greubel, Nov 12 2019 *)
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PROG
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(PARI) vector(20, n, prod(j=1, n-1, 2*j+7) ) \\ G. C. Greubel, Nov 12 2019
(MAGMA) [1] cat [(&*[2*j+9: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Nov 12 2019
(Sage) [product( (2*j+9) for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Nov 12 2019
(GAP) List([0..20], n-> Product([0..n-1], j-> 2*j+9) ); # G. C. Greubel, Nov 12 2019
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CROSSREFS
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Cf. A000165, A001147(n+1), A002866(n+1).
Cf. A051577, A051578, A051579, A051580 (rows m=0..6), A051582, A051583.
Sequence in context: A233907 A294120 A015685 * A080291 A215243 A113395
Adjacent sequences: A051578 A051579 A051580 * A051582 A051583 A051584
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KEYWORD
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easy,nonn
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AUTHOR
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Wolfdieter Lang
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STATUS
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approved
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