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A284441 Base-3 generalized Pascal triangle P_3 read by rows (see Comments for precise definition).

%I A284441 #26 Aug 26 2018 20:16:00
%S A284441 1,1,1,1,0,1,1,1,0,1,1,2,0,0,1,1,1,1,0,0,1,1,0,1,0,0,0,1,1,1,1,0,0,0,
%T A284441 0,1,1,0,2,0,0,0,0,0,1,1,1,0,2,0,0,0,0,0,1,1,2,0,1,1,0,0,0,0,0,1,1,1,
%U A284441 1,1,0,1,0,0,0,0,0,1,1,2,0,2,1,0,0,0,0,0,0,0,1,1,3,0,0,3,0,0,0,0,0,0,0,0,1,1,2,1,0,1,2,0,0,0,0,0,0,0,0,1,1
%N A284441 Base-3 generalized Pascal triangle P_3 read by rows (see Comments for precise definition).
%C A284441 List the base-3 numbers in their natural order as base-3 strings, beginning with the empty string epsilon, which represents 0. Row n of the triangle gives the number of times the k-th string occurs as a (scattered) substring of the n-th string.
%H A284441 Manon Stipulanti, <a href="/A284441/b284441.txt">Table of n, a(n) for n = 0..29645</a>
%H A284441 J. Leroy, M. Rigo, and M. Stipulanti, <a href="http://dx.doi.org/10.1016/j.disc.2017.01.003">Counting the number of non-zero coefficients in rows of generalized Pascal triangles</a>, Discrete Mathematics 340 (2017), 862-881.
%H A284441 Julien Leroy, Michel Rigo, Manon Stipulanti, <a href="https://arxiv.org/abs/1705.10065">Counting Subwords Occurrences in Base-b Expansions</a>, arXiv:1705.10065 [math.CO], 2017.
%H A284441 Julien Leroy, Michel Rigo, Manon Stipulanti, <a href="http://math.colgate.edu/~integers/sjs13/sjs13.Abstract.html">Counting Subwords Occurrences in Base-b Expansions</a>, Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A13.
%H A284441 Manon Stipulanti, <a href="https://arxiv.org/abs/1801.03287">Convergence of Pascal-Like Triangles in Parry-Bertrand Numeration Systems</a>, arXiv:1801.03287 [math.CO], 2018.
%e A284441 Triangle begins:
%e A284441 1,
%e A284441 1, 1,
%e A284441 1, 0, 1,
%e A284441 1, 1, 0, 1,
%e A284441 1, 2, 0, 0, 1,
%e A284441 1, 1, 1, 0, 0, 1,
%e A284441 1, 0, 1, 0, 0, 0, 1
%e A284441 1, 1, 1, 0, 0, 0, 0, 1,
%e A284441 1, 0, 2, 0, 0, 0, 0, 0, 1,
%e A284441 1, 1, 0, 2, 0, 0, 0, 0, 0, 1,
%e A284441 1, 2, 0, 1, 1, 0, 0, 0, 0, 0, 1,
%e A284441 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1,
%e A284441 1, 2, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1,
%e A284441 1, 3, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 1
%e A284441 ...
%e A284441 The base-3 numbers are epsilon, 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102, 110, 111, 112, 120, 121, 122, 200, 201, 202, 210, 211, 212, 220, 221, 222, ...  The tenth number 101 contains
%e A284441 eps 1 2 10 11 12 20 21 22 100 101 respectively
%e A284441 .1..2.0..1..1..0..0..0..0..0...1 times, which is row 10 of the triangle.
%t A284441 coeff[u_, v_] := coeff[u, v] = If[Length[v] == 0, 1, If[Length[u] < Length[v], 0, coeff[Drop[u, -1], v] + ((Last[u] == Last[v]) /. {True -> 1, False -> 0}) coeff[Drop[u, -1], Drop[v, -1]]]]
%t A284441 P3 = Table[coeff[IntegerDigits[i, 3] /. {0} -> {},IntegerDigits[j, 3] /. {0} -> {}], {i, 0, 3^5 - 1}, {j, 0, i}] //Flatten
%Y A284441 Cf. A282715 (gives (essentially) the number of nonzero entries in the rows), A284442 (their partial sums).
%K A284441 nonn,tabl
%O A284441 0,12
%A A284441 _Manon Stipulanti_, Mar 27 2017