The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A106271 Row sums of number triangle A106270. 2
1, 0, -2, -7, -21, -63, -195, -624, -2054, -6916, -23712, -82498, -290510, -1033410, -3707850, -13402695, -48760365, -178405155, -656043855, -2423307045, -8987427465, -33453694485, -124936258125, -467995871775, -1757900019099, -6619846420551, -24987199492703 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
From Petros Hadjicostas, Jul 15 2019: (Start)
To prove R. J. Mathar's conjecture, let A(x) be the g.f. of the current sequence. We note first that
Sum_{n >= 2} (n+1)*a(n)*x^n = (x*A(x))' - 1,
Sum_{n >= 2} (1-5*n)*a(n-1)*x^n = x*A(x) - 5*x*(x*A(x))' + 4*x, and
Sum_{n >= 2} 2*(2*n-1)*a(n-2)*x^n = 4*x*(x^2*A(x))' - 2*x^2*A(x).
Adding these equations (side by side), we get
Sum_{n >= 2} ((n+1)*a(n) + (1-5*n)*a(n-1) + 2*(2*n-1)*a(n-2))*x^n = 0,
which proves the conjecture. (End)
LINKS
FORMULA
G.f.: c(x)*sqrt(1 - 4x)/(1 - x), where c(x) is the g.f. of A000108.
a(n) = Sum_{k = 0..n} 2*0^(n-k) - C(n-k), where C(m) = A000108(m) (Catalan numbers).
a(n) = 2 - A014137(n) for n >= 0 and a(n) = 1 - A014138(n) for n >= 0. - Alexander Adamchuk, Feb 23 2007, corrected by Vaclav Kotesovec, Jul 22 2019
Conjecture: (n+1)*a(n) + (1-5*n)*a(n-1) + 2*(2*n-1)*a(n-2) = 0. - R. J. Mathar, Nov 09 2012
a(n) ~ -2^(2*n + 2) / (3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jul 22 2019
MATHEMATICA
Table[1 - Sum[(2n)!/n!/(n+1)!, {n, 1, k}], {k, 0, 30}] (* Alexander Adamchuk, Feb 23 2007 *)
CROSSREFS
Cf. A014138, A014137 (partial sums of Catalan numbers (A000108)).
Cf. A106270.
Sequence in context: A353094 A291411 A159972 * A027990 A037520 A218836
KEYWORD
easy,sign
AUTHOR
Paul Barry, Apr 28 2005
EXTENSIONS
More terms from Alexander Adamchuk, Feb 23 2007
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 14 14:51 EDT 2024. Contains 373400 sequences. (Running on oeis4.)