A003099 a(n) = Sum_{k=0..n} binomial(n,k^2).

1, 2, 3, 4, 6, 11, 22, 43, 79, 137, 231, 397, 728, 1444, 3018, 6386, 13278, 26725, 51852, 97243, 177671, 320286, 579371, 1071226, 2053626, 4098627, 8451288, 17742649, 37352435, 77926452, 159899767, 321468048, 632531039, 1219295320, 2308910353, 4314168202

Offset: 0

Author

N. J. A. Sloane, Henry W. Gould

Keywords

• nonn
• easy

References

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..3000
• Henry W. Gould, Fibonomial Catalan numbers: arithmetic properties and a table of the first fifty numbers, Abstract 71T-A216, Notices Amer. Math. Soc, 1971, page 938. [Annotated scanned copy of abstract]
• Henry W. Gould, Letter to N. J. A. Sloane, Nov 1973, and various attachments.
• Henry W. Gould, Letters to N. J. A. Sloane, Oct 1973 and Jan 1974.

Crossrefs

Cf. A010052, A206849.

Partial sums of A103198.

Programs

Magma

[(&+[Binomial(n, j^2): j in [0..n]]): n in [0..50]]; // G. C. Greubel, Oct 26 2022

Mathematica

Table[Sum[Binomial[n, k^2], {k, 0, Sqrt[n]}], {n, 0, 50}] (* T. D. Noe, Sep 10 2011 *)

PARI

a(n)=sum(k=0,sqrtint(n),binomial(n,k^2)) \\ Charles R Greathouse IV, Mar 26 2013

SageMath

def A003099(n): return sum( binomial(n,k^2) for k in range(isqrt(n)+1))
[A003099(n) for n in range(50)] # G. C. Greubel, Oct 26 2022

Formula

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]

Binomial transform of the characteristic function of squares A010052. - Carl Najafi, Sep 09 2011

G.f.: (1/(1 - x)) * Sum_{k>=0} (x/(1 - x))^(k^2). - Ilya Gutkovskiy, Jan 22 2024

Extensions

More terms from Carl Najafi, Sep 09 2011