A007179 Dual pairs of integrals arising from reflection coefficients. (Formerly M3284)

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

0,4

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).

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

Example

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)

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 *)

Prog

(Magma) [Floor(2^n/2-2^(n/2)*(1+(-1)^n)/4): n in [0..40]]; // Vincenzo Librandi, Aug 20 2011
(PARI) Vec(x*(1-x)/((1-2*x)*(1-2*x^2)) + O(x^50)) \\ Michel Marcus, Jan 28 2016

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.

Sequence in context: A059736 A261682 A102731 * A112576 A174804 A081487

Adjacent sequences: A007176 A007177 A007178 * A007180 A007181 A007182

Keyword

nonn,easy

Author

N. J. A. Sloane, Simon Plouffe

Status

approved