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%I M0576 #55 Jan 22 2024 16:09:23
%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} binomial(n,k^2).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A003099/b003099.txt">Table of n, a(n) for n = 0..3000</a>
%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 [These constants are sqrt(2/Pi) * JacobiTheta3(0,exp(-4)) = 0.827112271364145742... and sqrt(2/Pi) * JacobiTheta2(0,exp(-4)) = 0.587247586271786487... - _Vaclav Kotesovec_, Jan 15 2023]
%F Binomial transform of the characteristic function of squares A010052. - _Carl Najafi_, Sep 09 2011
%F G.f.: (1/(1 - x)) * Sum_{k>=0} (x/(1 - x))^(k^2). - _Ilya Gutkovskiy_, Jan 22 2024
%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
%o (Magma) [(&+[Binomial(n, j^2): j in [0..n]]): n in [0..50]]; // _G. C. Greubel_, Oct 26 2022
%o (SageMath)
%o def A003099(n): return sum( binomial(n,k^2) for k in range(isqrt(n)+1))
%o [A003099(n) for n in range(50)] # _G. C. Greubel_, Oct 26 2022
%Y Cf. A010052, A206849.
%Y Partial sums of A103198.
%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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