A007179 Dual pairs of integrals arising from reflection coefficients.

0, 1, 1, 4, 6, 16, 28, 64, 120, 256, 496, 1024, 2016, 4096, 8128, 16384, 32640, 65536, 130816, 262144, 523776, 1048576, 2096128, 4194304, 8386560, 16777216, 33550336, 67108864, 134209536, 268435456, 536854528, 1073741824, 2147450880, 4294967296, 8589869056

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy

Examples

			From _Gus Wiseman_, Feb 26 2022: (Start)
Also the number of integer compositions of n with at least one odd part. For example, the a(1) = 1 through a(5) = 16 compositions are:
  (1)  (1,1)  (3)      (1,3)      (5)
              (1,2)    (3,1)      (1,4)
              (2,1)    (1,1,2)    (2,3)
              (1,1,1)  (1,2,1)    (3,2)
                       (2,1,1)    (4,1)
                       (1,1,1,1)  (1,1,3)
                                  (1,2,2)
                                  (1,3,1)
                                  (2,1,2)
                                  (2,2,1)
                                  (3,1,1)
                                  (1,1,1,2)
                                  (1,1,2,1)
                                  (1,2,1,1)
                                  (2,1,1,1)
                                  (1,1,1,1,1)
(End)
		

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
• J. Heading, Theorem relating to the development of a reflection coefficient in terms of a small parameter, J. Phys. A 14 (1981), 357-367.
• Kyu-Hwan Lee, Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
• A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264 | doi:10.1246/bcsj.20140204. See Tables 1 and 2 (and text). - _N. J. A. Sloane_, Mar 26 2015
• Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Crossrefs

Column k=2 of A309748.

Odd bisection is A000302.

Even bisection is A006516 = 2^(n-1)*(2^n - 1).

The complement is counted by A077957, internal version A027383.

The internal case is A274230, even bisection A134057.

A000045(n-1) counts compositions without odd parts, non-singleton A077896.

A003242 counts Carlitz compositions.

A011782 counts compositions.

A034871, A097805, and A345197 count compositions by alternating sum.

A052952 (or A074331) counts non-singleton compositions without even parts.

Cf. A000918, A033484, A052955, A060867, A116406, A138364.

Programs

Magma

[Floor(2^n/2-2^(n/2)*(1+(-1)^n)/4): n in [0..40]]; // Vincenzo Librandi, Aug 20 2011

Maple

f := n-> if n mod 2 = 0 then 2^(n-1)-2^((n-2)/2) else 2^(n-1); fi;

Mathematica

LinearRecurrence[{2,2,-4},{0,1,1},30] (* Harvey P. Dale, Nov 30 2015 *)
Table[2^(n-1)-If[EvenQ[n],2^(n/2-1),0],{n,0,15}] (* Gus Wiseman, Feb 26 2022 *)

PARI

Vec(x*(1-x)/((1-2*x)*(1-2*x^2)) + O(x^50)) \\ Michel Marcus, Jan 28 2016

Formula

From Paul Barry, Apr 28 2004: (Start)

Binomial transform is (A000244(n)+A001333(n))/2.

G.f.: x*(1-x)/((1-2*x)*(1-2*x^2)).

a(n) = 2*a(n-1)+2*a(n-2)-4*a(n-3).

a(n) = 2^n/2-2^(n/2)*(1+(-1)^n)/4. (End)

G.f.: (1+x*Q(0))*x/(1-x), where Q(k)= 1 - 1/(2^k - 2*x*2^(2*k)/(2*x*2^k - 1/(1 + 1/(2*2^k - 8*x*2^(2*k)/(4*x*2^k + 1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 22 2013

a(n) = A011782(n+2) - A077957(n) - Gus Wiseman, Feb 26 2022