

A141017


List of largest row numbers of Pascallike triangles with index of asymmetry y = 1 and index of obliqueness z = 0 or z = 1.


0



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
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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, k1) + 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, k1) + G(n+1, k2) + G(n+2, k1) 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(n1, k1), 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 nth 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 * A190591 A332338 A332836
Adjacent sequences: A141014 A141015 A141016 * A141018 A141019 A141020


KEYWORD

nonn


AUTHOR

JuriStepan 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



