A141017 - OEIS

%I #42 Aug 25 2019 13:47:30

%S 1,1,2,4,7,12,23,46,89,168,311,594,1194,2355,4570,8745,16532,32948,

%T 65761,129632,252697,487647,936785,1884892,3754166,7407451,14489982,

%U 28118751,54868937,110096666,219129673,432847116,848952949,1654022768,3256427202,6524228863,12983131874,25671612977,50454577444

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

%C 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).

%C 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]

%C From _Petros Hadjicostas_, Jun 10 2019: (Start)

%C 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.

%C 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.

%C Thus, except for the (unfortunate) shifting of the indices by 1, triangular arrays A140998 and A140993 are mirror images of each other.

%C (End)

%F a(n) = max(A140993(n,k), k = 1..n). - _R. J. Mathar_, Apr 28 2010

%F a(n) = max(A140998(n-1, k-1), k = 1..n). - _Petros Hadjicostas_, Jun 10 2019

%e Triangle with y = 1 and z = 0 (i.e., triangle A140998) begins as follows:

%e a(1) = max(1) = 1;

%e a(2) = max(1, 1) = 1;

%e a(3) = max(1, 2, 1) = 2;

%e a(4) = max(1, 4, 2, 1) = 4;

%e a(5) = max(1, 7, 5, 2, 1) = 7;

%e a(6) = max(1, 12, 11, 5, 2, 1) = 12;

%e a(7) = max(1, 20, 23, 12, 5, 2, 1) = 23;

%e a(8) = max(1, 33, 46, 28, 12, 5, 2, 1) = 46;

%e a(9) = max(1, 54, 89, 63, 29, 12, 5, 2, 1) = 89;

%e ...

%p # Here, BB is the bivariate g.f. of sequence A140993.

%p 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;

%p #

%p # Here, we find the n-th row of sequence A140993 and find the maximum of the row:

%p ff := proc(n) local xx, k, yy;

%p xx := 0;

%p for k from 1 to n do

%p yy := coeftayl(coeftayl(BB(x, y), x = 0, n), y = 0, k);

%p xx := max(xx, yy); end do; xx;

%p end proc;

%p #

%p # Here, we print the maxima of the rows:

%p for i from 1 to 40 do

%p     ff(i);

%p end do; # _Petros Hadjicostas_, Jun 10 2019

%Y Cf. A007318, A140993, A140998.

%K nonn

%O 1,3

%A _Juri-Stepan Gerasimov_, Jul 11 2008

%E a(4) and offset corrected by _Gary W. Adamson_, Jul 11 2008

%E More terms from _R. J. Mathar_, Apr 28 2010

%E Name edited and more terms by _Petros Hadjicostas_, Jun 10 2019