|
|
A051579
|
|
a(n) = (2*n+5)!!/5!!, related to A001147 (odd double factorials).
|
|
10
|
|
|
1, 7, 63, 693, 9009, 135135, 2297295, 43648605, 916620705, 21082276215, 527056905375, 14230536445125, 412685556908625, 12793252264167375, 422177324717523375, 14776206365113318125, 546719635509192770625
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
Row m=5 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.
|
|
LINKS
|
|
|
FORMULA
|
a(n) = (2*n+5)!!/4!!.
E.g.f.: 1/(1-2*x)^(7/2).
a(n) ~ 8/15*sqrt(2)*n^3*2^n*e^-n*n^n*(1 + 107/24*n^-1 + ...). - Joe Keane (jgk(AT)jgk.org), Nov 23 2001
G.f.: G(0)/(10*x) -1/(5*x), where G(k)= 1 + 1/(1 - x*(2*k+5)/(x*(2*k+5) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 13 2013
a(n+1) = (2*n + 7)*a(n) with a(0) = 1.
O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 7*x)*A(x) - 1 with A(0) = 1.
G.f. as an S-fraction: A(x) = 1/(1 - 7*x/(1 - 2*x/(1 - 9*x/(1 - 4*x/(1 - 11*x/(1 - 6*x/(1 - ... - (2*n + 5)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 7*x/(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 - ...)))))))))). (End)
Sum_{n>=0} 1/a(n) = 15 * sqrt(e*Pi/2) * erf(1/sqrt(2)) - 20, where erf is the error function.
Sum_{n>=0} (-1)^n/a(n) = 15 * sqrt(Pi/(2*e)) * erfi(1/sqrt(2))) - 10, where erfi is the imaginary error function. (End)
|
|
MAPLE
|
df:=doublefactorial; seq(df(2*n+5)/df(5), n = 0..20); # G. C. Greubel, Nov 12 2019
|
|
MATHEMATICA
|
Table[2^n*Pochhammer[7/2, n], {n, 0, 20}] (* G. C. Greubel, Nov 12 2019 *)
|
|
PROG
|
(PARI) vector(20, n, prod(j=1, n-1, 2*j+5) ) \\ G. C. Greubel, Nov 12 2019
(Magma) [1] cat [(&*[2*j+7: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Nov 12 2019
(Sage) [product( (2*j+7) 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+7) ); # G. C. Greubel, Nov 12 2019
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|