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A076729
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Numerator of integral_{x=0..1} (1+x^2)^n dx.
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5
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1, 4, 28, 288, 3984, 70080, 1506240, 38384640, 1133072640, 38038533120, 1431213235200, 59645279232000, 2726781752217600, 135661078090137600, 7295806823277772800, 421717409630060544000, 26071235813929033728000
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OFFSET
| 0,2
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COMMENTS
| 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
a(n)=2*n*a(n-1)+(2*n)!/n!
a(n)=2^n*sum(A001147(k)*A001147(n-k),k=0..n)
a(n)=(2*n+1)!*sum(k!*(-2)^k/((2*k+1)!*(n-k)!),k=0..n)
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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
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EXAMPLE
| For n=3, (2n+1)!!=105 and the integral is 96/35 = 288/105, so a(3) = 288.
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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)
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PROG
| (PARI) a(n)=if(n<0, 0, subst(intformal((1+x^2)^n), x, 1)*(2*n+1)!/2^n/n!)
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CROSSREFS
| Cf. A077595, A077745, A086891.
Sequence in context: A071212 A090353 A201595 * A078634 A091485 A201354
Adjacent sequences: A076726 A076727 A076728 * A076730 A076731 A076732
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KEYWORD
| nonn
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AUTHOR
| Al Hakanson (hawku(AT)hotmail.com), Oct 28 2002
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