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A027306 a(n) = 2^(n-1) + (1 + (-1)^n)/4*binomial(n, n/2). 27
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

Zachary Hamaker, Eric Marberg, Atoms for signed permutations, arXiv:1802.09805 [math.CO], 2018.

Donatella Merlini, Massimo Nocentini, Algebraic Generating Functions for Languages Avoiding Riordan Patterns, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.3.

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 August 21 21:42 EDT 2018. Contains 313957 sequences. (Running on oeis4.)