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

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, 0, 1, 8, 33, 88, 173, 173, 1, 9, 42, 130, 303, 476, 0, 1, 10, 52, 182, 485, 961, 961, 0, 1, 11, 63, 245, 730, 1691, 2652, 2652, 1, 12, 75, 320, 1050, 2741, 5393, 8045, 0

Offset: 1

Author

Ruud H.G. van Tol, Dec 28 2023

Keywords

• nonn
• tabf

Examples

			Triangle T(n,k) begins:
 n|k:1|  2|  3|  4|  5|  6|  7|  8|...
--+---+---+---+---+---+---+---+---+---
 1|  1
 2|  1   0
 3|  1   1
 4|  1   2   0
 5|  1   3   3
 6|  1   4   7   0
 7|  1   5  12  12   0
 8|  1   6  18  30  30
 9|  1   7  25  55  85   0
10|  1   8  33  88 173 173
11|  1   9  42 130 303 476   0
12|  1  10  52 182 485 961 961   0
...
		

Links

• Ruud H.G. van Tol, Table of n, a(n) for n = 1..11858 (200 rows).
• Ruud H.G. van Tol, Sequence on a lattice

Crossrefs

Cf. A009766 (Catalan's triangle), A098294 (row lengths), A100982 (row sums).

Cf. A055998, A111396.

Programs

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

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

Formula

Row length L(n) = A098294(n) = floor(n*log(3)/log(2)) + 1 - n.

T(n,1) = 1.

T(n+1,k) = T(n+1,k-1) + T(n,k) for 1 < k <= L(n).

T(n+1,L(n+1)) = 0 if L(n+1) > L(n).

T(n+1,2) = n-1.

T(n+3,3) = A055998(n-1) = (n-1)*(n+4)/2.

T(n+5,4) = A111396(n-1) = (n-1)*(n+6)*(n+7)/6.

T(n+1,k) = Sum_{j=1..k} T(n,j) for 1 <= k <= L(n).

Extensions

Corrected by Ruud H.G. van Tol, Nov 29 2024