A141017 List of largest row numbers of Pascal-like triangles with index of asymmetry y = 1 and index of obliqueness z = 0 or z = 1.

1, 1, 2, 4, 7, 12, 23, 46, 89, 168, 311, 594, 1194, 2355, 4570, 8745, 16532, 32948, 65761, 129632, 252697, 487647, 936785, 1884892, 3754166, 7407451, 14489982, 28118751, 54868937, 110096666, 219129673, 432847116, 848952949, 1654022768, 3256427202, 6524228863, 12983131874, 25671612977, 50454577444

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)

Links

Table of n, a(n) for n=1..39.

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

Cf. A007318, A140993, A140998.

Sequence in context: A018080 A226160 A018181 * A357925 A190591 A332338

Adjacent sequences: A141014 A141015 A141016 * A141018 A141019 A141020

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