A073028 a(n) = max{ C(n,0), C(n-1,1), C(n-2,2), ..., C(n-n,n) }.

1, 1, 1, 2, 3, 4, 6, 10, 15, 21, 35, 56, 84, 126, 210, 330, 495, 792, 1287, 2002, 3003, 5005, 8008, 12376, 19448, 31824, 50388, 77520, 125970, 203490, 319770, 497420, 817190, 1307504, 2042975, 3268760, 5311735, 8436285, 13123110, 21474180, 34597290

Offset: 0

Author

Miklos Kristof, Aug 22 2002

Keywords

• easy
• nonn

Comments

lim a(n)/a(n-1) = (1+sqrt(5))/2.

a(n-1) is the max coefficient in n-th Fibonacci polynomial (the polynomial F_0(x) is constant zero, and is not included in this sequence). - Vladimir Reshetnikov, Oct 09 2016

Examples

			For n = 6, C(6,0) = 1, C(5,1) = 5, C(4,2) = 6, C(3,3) = 1. These binomial coefficients are the coefficients in the Fibonacci polynomial F_7(x) = x^6 + 5*x^4 + 6*x^2 + 1. The max coefficient is 6, so a(6) = 6.
		

References

• Peter Boros (borospet(AT)freemail.hu): Lectures on Fibonacci's World at the SOTERIA Foundation, 1999.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 0..4793
• Benjamin Aram Berendsohn, László Kozma, and Dániel Marx, Finding and counting permutations via CSPs, arXiv:1908.04673 [cs.DS], 2019.
• Charles Bouillaguet, Boolean Polynomial Evaluation for the Masses, LIP6 Laboratory, Sorbonne Université (Paris, France) Cryptology ePrint Archive (2022) No. 1412.
• S. M. Tanny and M. Zuker, On a unimodal sequence of binomial coefficients, Discrete Math. 9 (1974), 79-89.
• Eric Weisstein's World of Mathematics, Fibonacci Polynomial.

Crossrefs

Cf. A060065, A277282, A168561.

Programs

Mathematica

Table[Max[CoefficientList[Fibonacci[n + 1, x], x]], {n, 1, 30}] (* Vladimir Reshetnikov, Oct 07 2016 *)

PARI

a(n)=my(k=(5*n-sqrtint(5*n^2+10*n+9)+6)\10); binomial(n-k,k) \\ Charles R Greathouse IV, Sep 22 2016

Formula

a(n) = binomial(n-A060065(n), A060065(n)). - Vladeta Jovovic, Jun 16 2004

a(n) ~ 5^(1/4) * phi^(n+1) / sqrt(2*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 09 2016

Extensions

a(0) = 1 prepended by Vladimir Reshetnikov, Oct 09 2016