|
Search: id:A076729
|
|
|
| A076729 |
|
Numerator of integral_{x=0..1} (1+x^2)^n dx. |
|
+0
6
|
|
| 1, 4, 28, 288, 3984, 70080, 1506240, 38384640, 1133072640, 38038533120, 1431213235200, 59645279232000, 2726781752217600, 135661078090137600, 7295806823277772800, 421717409630060544000, 26071235813929033728000
(list; graph; listen)
|
|
|
OFFSET
|
0,2
|
|
|
COMMENT
|
Denominator is equal to (2n+1)!! = A001147(n+1).
Also numerator of the integral (1-x^2)^-(n+.5) for x from 0 to sqrt 1/2. Here the sequence starts at n=1; at n=2 the function is 4.
a(n)=(integral_{x=0 to ln(1+sqrt2)} cosh(x)^(2*n-1) dx) where the denominators are b(n)=(2*n)!/(n!*2^n). E.g. a(3)= 28 and b(3)= 15. both offsets are 1. - Al Hakanson (hawkuu(AT)excite.com), Mar 02 2004
|
|
FORMULA
|
(2*n+1)!!*hypergeom([1/2, -n], [3/2], -1). - Vladeta Jovovic (vladeta(AT)eunet.rs), Dec 05 2002
E.g.f.: 1/((1-2*x)*sqrt(1-4*x)). - Vladeta Jovovic (vladeta(AT)eunet.rs), May 11 2003
|
|
EXAMPLE
|
For n=3, (2n+1)!!=105 and the integral is 96/35 = 288/105, so a(3) = 288.
|
|
MATHEMATICA
|
f[n_] := (2n + 1)!!*Integrate[(1 + x^2)^n, {x, 0, 1}]; Table[ f[n], {n, 0, 16}] (from Robert G. Wilson v Feb 27 2004)
|
|
PROGRAM
|
(PARI) a(n)=if(n<0, 0, subst(intformal((1+x^2)^n), x, 1)*(2*n+1)!/2^n/n!)
|
|
CROSSREFS
|
Cf. A077595, A077745, A086891.
Sequence in context: A138208 A071212 A090353 this_sequence A078634 A091485 A112938
Adjacent sequences: A076726 A076727 A076728 this_sequence A076730 A076731 A076732
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Al Hakanson (hawku(AT)hotmail.com), Oct 28 2002
|
|
|
Search completed in 0.002 seconds
|
