OFFSET
1,3
COMMENTS
Triangle with index of asymmetry y = 1 and index of obliqueness z = 0, read by rows, with recurrence for G(n, k) as follows: G(n, 0) = G(n+1, n+1) = 1, G(n+2, n+1) = 2, G(n+3, k) = G(n+1, k-1) + G(n+1, k) + G(n+2, k) for k = 1..(n+1).
Triangle with index of asymmetry y = 1 and index of obliqueness z = 1, read by rows, with recurrence for G(n, k) as follows: G(n, n) = G(n+1, 0) = 1, G(n+2, 1) = 2, G(n+3, k) = G(n+1, k-1) + G(n+1, k-2) + G(n+2, k-1) for k = 2..(n+2). [Edited by Petros Hadjicostas, Jun 11 2019]
From Petros Hadjicostas, Jun 10 2019: (Start)
For the triangle with index of asymmetry y = 1 and index of obliqueness z = 0, read by rows, we have G(n, k) = A140998(n, k) for 0 <= k <= n.
For the triangle with index of asymmetry y = 1 and index of obliqueness z = 1, read by rows, we have G(n, k) = A140993(n+1, k+1) for n >= 0 and k >= 0.
Thus, except for the (unfortunate) shifting of the indices by 1, triangular arrays A140998 and A140993 are mirror images of each other.
(End)
FORMULA
a(n) = max(A140993(n,k), k = 1..n). - R. J. Mathar, Apr 28 2010
a(n) = max(A140998(n-1, k-1), k = 1..n). - Petros Hadjicostas, Jun 10 2019
EXAMPLE
Triangle with y = 1 and z = 0 (i.e., triangle A140998) begins as follows:
a(1) = max(1) = 1;
a(2) = max(1, 1) = 1;
a(3) = max(1, 2, 1) = 2;
a(4) = max(1, 4, 2, 1) = 4;
a(5) = max(1, 7, 5, 2, 1) = 7;
a(6) = max(1, 12, 11, 5, 2, 1) = 12;
a(7) = max(1, 20, 23, 12, 5, 2, 1) = 23;
a(8) = max(1, 33, 46, 28, 12, 5, 2, 1) = 46;
a(9) = max(1, 54, 89, 63, 29, 12, 5, 2, 1) = 89;
...
MAPLE
# Here, BB is the bivariate g.f. of sequence A140993.
BB := proc(x, y) y*x*(1 - y*x - x^2*y^2 + x^3*y^2)/((1 - x)*(1 - y*x)*(1 - y*x - x^2*y - x^2*y^2)); end proc;
#
# Here, we find the n-th row of sequence A140993 and find the maximum of the row:
ff := proc(n) local xx, k, yy;
xx := 0;
for k from 1 to n do
yy := coeftayl(coeftayl(BB(x, y), x = 0, n), y = 0, k);
xx := max(xx, yy); end do; xx;
end proc;
#
# Here, we print the maxima of the rows:
for i from 1 to 40 do
ff(i);
end do; # Petros Hadjicostas, Jun 10 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Juri-Stepan Gerasimov, Jul 11 2008
EXTENSIONS
a(4) and offset corrected by Gary W. Adamson, Jul 11 2008
More terms from R. J. Mathar, Apr 28 2010
Name edited and more terms by Petros Hadjicostas, Jun 10 2019
STATUS
approved