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A043301
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2^n*Sum_{ k=0..n } (n+k)!/((n-k)!*k!*4^k).
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6
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1, 3, 13, 77, 591, 5627, 64261, 857901, 13125559, 226566107, 4357258269, 92408688077, 2142828858847, 53940356223483, 1464960933469429, 42699628495507373, 1329548327094606279, 44045893308104036699
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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REFERENCES
| Bruce Berndt, Ramanujan's Notebooks Part II, Springer-Verlag; see Integrals and Asymptotic Expansions, p. 229.
I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and P roducts, 6th ed., Section 3.737.1, p. 423.
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FORMULA
| a(n)=(2*n-1)*a(n-1)+4*a(n-2), n>1.
2^(n+1)n!(e^2/Pi)*Integral(t=0, infinity, cos(2t)/(1+t^2)^(n+1)dt).
E.g.f.: 2*(e^2/Pi)*Integral(t=0, infinity, cos(2t)/(1+t^2-2x)dt).
2^n * y_n(1/2), where y_n(x) are the Bessel polynomials A001498.
G.f.: A(x) = 1/G(0) ; G(k) = 1 - 2*x - x*(k+1)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 17 2011
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MATHEMATICA
| Table[2^n Sum[(n+k)!/((n-k)!k! 4^k), {k, 0, n}], {n, 0, 20}] (* or *) RecurrenceTable[{a[0]==1, a[1]==3, a[n]==(2n-1)a[n-1]+4a[n-2]}, a[n], {n, 20}] (* From Harvey P. Dale, Aug 14 2011 *)
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CROSSREFS
| Cf. A043302, A144505.
Sequence in context: A159662 A032035 A127127 * A141762 A062872 A159312
Adjacent sequences: A043298 A043299 A043300 * A043302 A043303 A043304
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KEYWORD
| nonn,easy
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AUTHOR
| Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 04 2002
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EXTENSIONS
| Edited by Michael Somos, Jul 16 2002
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