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A027306 a(n) = 2^(n-1) + (1 + (-1)^n)/4*binomial(n, n/2). 18
1, 1, 3, 4, 11, 16, 42, 64, 163, 256, 638, 1024, 2510, 4096, 9908, 16384, 39203, 65536, 155382, 262144, 616666, 1048576, 2449868, 4194304, 9740686, 16777216, 38754732, 67108864, 154276028, 268435456, 614429672, 1073741824, 2448023843 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Inverse binomial transform of A027914. Hankel transform (see A001906 for definition) is {1, 2, 3, 4, ..., n, ...}. - Philippe Deléham, Jul 21 2005

Number of walks of length n on a line that starts at the origin and ends at or above 0. - Benjamin Phillabaum, Mar 05 2011

Number of binary integers (i.e., with a leading 1 bit) of length n+1 which have a majority of 1-bits. E.g., for n+1=4: (1011, 1101, 1110, 1111) a(3)=4. - Toby Gottfried, Dec 11 2011

REFERENCES

A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (4.2.1.6)

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

F. Disanto, A. Frosini, S. Rinaldie, Square involuations, J. Int. Seq. 14 (2011) # 11.3.5

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

FORMULA

a(n) = Sum_{k=0..[n/2]} C(n,k).

Odd terms are 2^(n-1). Also a(2n) - 2^(2n-1) is given by A001700. a(n) = 2^n+mod(n, 2)*C(n, (n-1)/2).

E.g.f.: (exp(2x)+I_0(2x))/2.

O.g.f.: 2*x/(1-2*x)/(1+2*x-((1+2*x)*(1-2*x))^(1/2)). - Vladeta Jovovic, Apr 27 2003

a(n) = A008949(n,floor(n/2)); a(n) + a(n-1) = A248574(n), n > 0. - Reinhard Zumkeller, Nov 14 2014

From Peter Bala, Jul 21 2015: (Start)

a(n) = [x^n]( 2*x - 1/(1 - x) )^n.

O.g.f. 1/2*( 1/sqrt(1 - 4*x^2) + 1/(1 - 2*x) ).

Inverse binomial transform is (-1)^n*A246437(n).

exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + ... is the o.g.f. for A001405. (End)

a(n) = Sum_{k=1..floor((n+1)/2)} binomial(n-1,(2n+1-(-1)^n)/4 -k). - Anthony Browne, Jun 18 2016

n*a(n) +2*(-n+1)*a(n-1) +4*(-n+1)*a(n-2) +8*(n-2)*a(n-3)=0. - R. J. Mathar, Aug 09 2017

MAPLE

a:= proc(n) option remember; if n=0 then 1 else add(binomial(n, j), j=0..n/2) fi end: seq(a(n), n=0..32); # Zerinvary Lajos, Mar 29 2009

MATHEMATICA

Table[Sum[Binomial[n, k], {k, 0, Floor[n/2]}], {n, 1, 35}]

(* Second program: *)

a[0] = a[1] = 1; a[2] = 3; a[n_] := a[n] = (2(n-1)(2a[n-2] + a[n-1]) - 8(n-2) a[n-3])/n; Array[a, 33, 0] (* Jean-François Alcover, Sep 04 2016 *)

PROG

(PARI) a(n)=if(n<0, 0, (2^n+if(n%2, 0, binomial(n, n/2)))/2)

(Haskell)

a027306 n = a008949 n (n `div` 2)  -- Reinhard Zumkeller, Nov 14 2014

(MAGMA) [2^(n-1)+(1+(-1)^n)/4*Binomial(n, n div 2): n in [0..40]]; // Vincenzo Librandi, Jun 19 2016

CROSSREFS

a(n) = Sum{(k+1)T(n, m-k)}, 0<=k<=[ (n+1)/2 ], T given by A008315.

Column k=2 of A226873. - Alois P. Heinz, Jun 21 2013

Cf. A008949, A248574, A001405, A246437.

Sequence in context: A001641 A007382 A127804 * A239024 A026676 A142870

Adjacent sequences:  A027303 A027304 A027305 * A027307 A027308 A027309

KEYWORD

nonn,easy,walk

AUTHOR

Clark Kimberling

EXTENSIONS

Better description from Robert G. Wilson v, Aug 30 2000 and from Yong Kong (ykong(AT)curagen.com), Dec 28 2000

STATUS

approved

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Last modified February 18 05:48 EST 2018. Contains 299298 sequences. (Running on oeis4.)