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A003099 a(n) = Sum_{k=0..n} C(n,k^2).
(Formerly M0576)

%I M0576

%S 1,2,3,4,6,11,22,43,79,137,231,397,728,1444,3018,6386,13278,26725,

%T 51852,97243,177671,320286,579371,1071226,2053626,4098627,8451288,

%U 17742649,37352435,77926452,159899767,321468048,632531039,1219295320,2308910353,4314168202

%N a(n) = Sum_{k=0..n} C(n,k^2).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Henry W. Gould, <a href="/A003099/a003099_2.pdf">Fibonomial Catalan numbers: arithmetic properties and a table of the first fifty numbers</a>, Abstract 71T-A216, Notices Amer. Math. Soc, 1971, page 938. [Annotated scanned copy of abstract]

%H Henry W. Gould, <a href="/A003099/a003099_1.pdf">Letter to N. J. A. Sloane, Nov 1973, and various attachments</a>.

%H Henry W. Gould, <a href="/A003099/a003099.pdf">Letters to N. J. A. Sloane, Oct 1973 and Jan 1974</a>.

%F a(n)*sqrt(n)/2^n is bounded: lim sup a(n)*sqrt(n)/2^n = 0.82... and lim inf a(n)*sqrt(n)/2^n = 0.58... - _Benoit Cloitre_, Nov 14 2003

%F Binomial transform of the characteristic function of squares A010052. - _Carl Najafi_, Sep 09 2011

%t Table[Sum[Binomial[n, k^2], {k, 0, Sqrt[n]}], {n, 0, 50}] (* _T. D. Noe_, Sep 10 2011 *)

%o (PARI) a(n)=sum(k=0,sqrtint(n),binomial(n,k^2)) \\ _Charles R Greathouse IV_, Mar 26 2013

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, _Henry W. Gould_

%E More terms from _Carl Najafi_, Sep 09 2011

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Last modified March 21 07:23 EDT 2019. Contains 321367 sequences. (Running on oeis4.)