Search: id:A136444
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%I A136444
%S A136444 0,0,0,1,3,6,12,25,51,101,197,381,731,1392,2634,4958,9290,17337,32239,
               59760,
%T A136444 110460,203651,374593,687567,1259597,2303449,4205493,7666560,13956532,
               25374108,
%U A136444 46076436,83575025,151431099,274108826,495708364,895670733,1617003823,
               2916984121
%N A136444 Sum k*binomial(n-k, 2k).
%C A136444 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}.
%C A136444 Sequence A_n, with generating function (1-z)/p(z) where p(z)=1-2z+z^2-z^3, 
               is A005251.
%C A136444 Sequence B_n, with generating function z/p(z), is A005314.
%C A136444 Sequence A^*_n is the present sequence.
%C A136444 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.
%C A136444 These sequences have many inter-relations; for example
%C A136444 B_{n+1}-B_n = A_n; B^*_{n+1}-B^*_n = A^*_n;
%C A136444 A_{n+1}-A_n = B_{n-1}; A^*_{n+1}-A^*_n = B^*_{n-1}+B_{n-1}.
%D A136444 D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.
%F A136444 G.f.: z^3*(1-z)/(1-2*z+z^2-z^3)^2.
%p A136444 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]
%Y A136444 Cf. A005251, A005314, A118430, A136445, A137356-A137361.
%Y A136444 Sequence in context: A007239 A088970 A068425 this_sequence A077854 A099445 
               A004067
%Y A136444 Adjacent sequences: A136441 A136442 A136443 this_sequence A136445 A136446 
               A136447
%K A136444 nonn
%O A136444 0,5
%A A136444 D. E. Knuth, Apr 04 2008

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