A110914 "Self-convolution mod 3" of central Delannoy numbers (see comment).

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, 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, 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

Offset

0,3

Comments

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.

Links

Table of n, a(n) for n=0..100.

E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, arXiv:math/0407326 [math.CO], 2004.

E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215.

Formula

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).

Mathematica

b[n_] := Sum[Binomial[n, k] Binomial[n + k, k], {k, 0, n}];
a[n_] := Sum[Mod[b[k] b[n - k], 3], {k, 0, n}];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 17 2019 *)

Prog

(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)

Crossrefs

Cf. A062756.

Sequence in context: A158945 A156667 A178090 * A219200 A341978 A193527

Adjacent sequences: A110911 A110912 A110913 * A110915 A110916 A110917

Keyword

nonn

Author

Benoit Cloitre, Oct 04 2005

Status

approved