0,5

Consider four related sequences: A_n = \sum {n-k \choose 2k}, B_n = \sum {n-k \choose 2k+1}, A^*_n = \sum k{n-k \choose 2k}, B^*_n = \sum k{n-k \choose 2k+1}.

Sequence A_n, with generating function (1-z)/p(z) where p(z)=1-2z+z^2-z^3, is A005251.

Sequence B_n, with generating function z/p(z), is A005314.

Sequence A^*_n is the present sequence.

Sequence B^*_n is A118430, but shifted one place so that the generating function is z^4/p(z)^2 instead of z^3/p(z)^2.

These sequences have many inter-relations; for example

B_{n+1}-B_n = A_n; B^*_{n+1}-B^*_n = A^*_n;

A_{n+1}-A_n = B_{n-1}; A^*_{n+1}-A^*_n = B^*_{n-1}+B_{n-1}.

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

G.f.: z^3*(1-z)/(1-2*z+z^2-z^3)^2.

a := n-> (Matrix([[0, 0, 1, 1, -3, -5]]). Matrix(6, (i, j)-> if (i=j-1) then 1 elif j=1 then [4, -6, 6, -5, 2, -1][i] else 0 fi)^n)[1, 1]; seq (a(n), n=0..37); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 13 2008]

Cf. A005251, A005314, A118430, A136445, A137356-A137361.

Adjacent sequences: A136441 A136442 A136443 this_sequence A136445 A136446 A136447

Sequence in context: A007239 A088970 A068425 this_sequence A077854 A099445 A004067

nonn

D. E. Knuth, Apr 04 2008