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A368514 Irregular triangle T(n,k) read by rows: similar to A009766 but length of rows grows like log(3)/log(2).

%I A368514 #52 Dec 12 2024 23:10:07
%S A368514 1,1,0,1,1,1,2,0,1,3,3,1,4,7,0,1,5,12,12,0,1,6,18,30,30,1,7,25,55,85,
%T A368514 0,1,8,33,88,173,173,1,9,42,130,303,476,0,1,10,52,182,485,961,961,0,1,
%U A368514 11,63,245,730,1691,2652,2652,1,12,75,320,1050,2741,5393,8045,0
%N A368514 Irregular triangle T(n,k) read by rows: similar to A009766 but length of rows grows like log(3)/log(2).
%H A368514 Ruud H.G. van Tol, <a href="/A368514/b368514.txt">Table of n, a(n) for n = 1..11858</a> (200 rows).
%H A368514 Ruud H.G. van Tol, <a href="/A368514/a368514.svg">Sequence on a lattice</a>
%F A368514 Row length L(n) = A098294(n) = floor(n*log(3)/log(2)) + 1 - n.
%F A368514 T(n,1) = 1.
%F A368514 T(n+1,k) = T(n+1,k-1) + T(n,k) for 1 < k <= L(n).
%F A368514 T(n+1,L(n+1)) = 0 if L(n+1) > L(n).
%F A368514 T(n+1,2) = n-1.
%F A368514 T(n+3,3) = A055998(n-1) = (n-1)*(n+4)/2.
%F A368514 T(n+5,4) = A111396(n-1) = (n-1)*(n+6)*(n+7)/6.
%F A368514 T(n+1,k) = Sum_{j=1..k} T(n,j) for 1 <= k <= L(n).
%e A368514 Triangle T(n,k) begins:
%e A368514  n|k:1|  2|  3|  4|  5|  6|  7|  8|...
%e A368514 --+---+---+---+---+---+---+---+---+---
%e A368514  1|  1
%e A368514  2|  1   0
%e A368514  3|  1   1
%e A368514  4|  1   2   0
%e A368514  5|  1   3   3
%e A368514  6|  1   4   7   0
%e A368514  7|  1   5  12  12   0
%e A368514  8|  1   6  18  30  30
%e A368514  9|  1   7  25  55  85   0
%e A368514 10|  1   8  33  88 173 173
%e A368514 11|  1   9  42 130 303 476   0
%e A368514 12|  1  10  52 182 485 961 961   0
%e A368514 ...
%o A368514 (PARI) row(n) = my(v=Vec([1], logint(3^n, 2)+1-n), c=1); for(i=2, n, for(j=2, c, v[j]+=v[j-1]); c=logint(3^i,2)+1-i); v
%o A368514 (PARI) rows(n) = my(v=vector(n, i, Vec([1], logint(3^i,2)+1-i))); for(i=3, n, for(j=2, #v[i-1], v[i][j]=v[i][j-1]+v[i-1][j])); v
%Y A368514 Cf. A009766 (Catalan's triangle), A098294 (row lengths), A100982 (row sums).
%Y A368514 Cf. A055998, A111396.
%K A368514 nonn,tabf
%O A368514 1,7
%A A368514 _Ruud H.G. van Tol_, Dec 28 2023
%E A368514 Corrected by _Ruud H.G. van Tol_, Nov 29 2024