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A110914 "Self convolution mod 3" of central Delannoy numbers (see comment). 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, 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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified May 25 14:51 EDT 2022. Contains 354071 sequences. (Running on oeis4.)