A049610 a(n) = Sum_{k=0..floor(n/2)} k*binomial(n,2*k) = floor(n*2^(n-3)).

0, 0, 1, 3, 8, 20, 48, 112, 256, 576, 1280, 2816, 6144, 13312, 28672, 61440, 131072, 278528, 589824, 1245184, 2621440, 5505024, 11534336, 24117248, 50331648, 104857600, 218103808, 452984832, 939524096, 1946157056, 4026531840, 8321499136, 17179869184, 35433480192

Offset: 0

Author

M. F. Hasler, Jan 25 2012

Keywords

• nonn
• easy

Comments

Essentially same as A001792, except for leading zeros, which motivate the existence of this sequence on its own.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Silvana Ramaj, New Results on Cyclic Compositions and Multicompositions, Master's Thesis, Georgia Southern Univ., 2021. See p. 67.
• Index entries for linear recurrences with constant coefficients, signature (4,-4).

Crossrefs

Cf. A001792, A189162, A189390, A189391.

Programs

Mathematica

CoefficientList[Series[x^2*(1 - x)/(1 - 2*x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Jan 09 2013 *)

PARI

a(n)=n<<(n-3)

Formula

G.f. x^2*(1-x)/(1-2*x)^2. - Sergei N. Gladkovskii, Oct 18 2012

G.f.: x^2*( 1 + 2*x*U(0) ) where U(k) = 1 + (k+1)/(2 - 8*x/(4*x + (k+1)/U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 19 2012

E.g.f.: x*(exp(2*x) - 1)/4. - Stefano Spezia, Feb 02 2023

Sum_{n>=2} 1/a(n) = 8*log(2) - 4. - Amiram Eldar, Feb 14 2023