A110914 - OEIS

%I #18 Aug 24 2022 03:04:45

%S 1,0,2,0,1,0,2,0,4,0,2,0,1,0,2,0,1,0,2,0,4,0,2,0,4,0,8,0,4,0,2,0,4,0,

%T 2,0,1,0,2,0,1,0,2,0,4,0,2,0,1,0,2,0,1,0,2,0,4,0,2,0,4,0,8,0,4,0,2,0,

%U 4,0,2,0,4,0,8,0,4,0,8,0,16,0,8,0,4,0,8,0,4,0,2,0,4,0,2,0,4,0,8,0,4

%N "Self-convolution mod 3" of central Delannoy numbers (see comment).

%C a(n) = Sum_{k=0..n} ((b(k)*b(n-k)) mod 3) where b(k) = Sum_{k=0..n} binomial(n,k)*binomial(n+k,k) are the central Delannoy numbers. The formula is obtained using techniques described in the Deutsch-Sagan paper.

%H E. Deutsch and B. E. Sagan, <a href="https://arxiv.org/abs/math/0407326">Congruences for Catalan and Motzkin numbers and related sequences</a>, arXiv:math/0407326 [math.CO], 2004.

%H E. Deutsch and B. E. Sagan, <a href="https://doi.org/10.1016/j.jnt.2005.06.005">Congruences for Catalan and Motzkin numbers and related sequences</a>, J. Num. Theory 117 (2006), 191-215.

%F a(2n-1)=0 and a(2n)=2^t_1(n) where t_1(n) denotes the number of 1's in the ternary representation of n (A062756). Recurrence: a(3n)=a(n), a(3n+1)=a(n-1), a(3n+2)=2*a(n).

%t b[n_] := Sum[Binomial[n, k] Binomial[n + k, k], {k, 0, n}];

%t a[n_] := Sum[Mod[b[k] b[n - k], 3], {k, 0, n}];

%t Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Feb 17 2019 *)

%o (PARI) b(n)=sum(k=0,n,binomial(n,k)*binomial(n+k,k)); a(n)=sum(k=0,n,(b(k)*b(n-k))%3)

%Y Cf. A062756.

%K nonn

%O 0,3

%A _Benoit Cloitre_, Oct 04 2005