OFFSET
1,3
COMMENTS
For the Pascal-like triangle G(n, k) with index of asymmetry y = 3 and index of obliqueness z = 0, which is read by rows, we have G(n, 0) = G(n+1, n+1) = 1, G(n+2, n+1) = 2, G(n+3, n+1) = 4, G(n+4, n+1) = 8, and G(n+5, k) = G(n+1, k-1) + G(n+1, k) + G(n+2, k) + G(n+3, k) + G(n+4, k) for n >= 0 and k = 1..(n+1). (This is array A140996.)
For the Pascal-like triangle G(n, k) with index of asymmetry y = 3 and index of obliqueness z = 1, which is read by rows, we have G(n, n) = G(n+1, 0) = 1, G(n+2, 1) = 2, G(n+3, 2) = 4, G(n+4, 3) = 8, and G(n+5, k) = G(n+1, k-3) + G(n+1, k-4) + G(n+2, k-3) + G(n+3, k-2) + G(n+4, k-1) for n >= 0 and for k = 4..(n+4). (This is array A140995.)
From Petros Hadjicostas, Jun 13 2019: (Start)
In the example below the author uses array A140996 to create this sequence. If we use array A140995, which is the mirror image of A140996, and follow the same process, we get a different sequence: 1, 1, 2, 3, 4, 7, 8, 15, 16, 31, 33, 63, 68, 127, 140, 255, 288, 512, 592, ...
Even though array A140996 starts at row n = 0, the offset of the current sequence was set at n = 1. In other words, a(n) = Sum_{ceiling((n-1)/2) <= i <= n-1} G(i, n-1-i) = G(n-1, 0) + G(n-2, 1) + ... + G(ceiling((n-1)/2), floor((n-1)/2)) for n >= 1, where G(n, k) = A140996(n, k).
To get the g.f. of this sequence, we take the bivariate g.f. of sequence A140996, and set x = y. We multiply the result by x because the offset here was set at n = 1.
Finally, we mention that in the attached photograph about Stepan's triangle, the index of asymmetry is denoted by s (rather than y) and the index of obliqueness is denoted by e (rather than z). For the Pascal triangle, s = y = 0.
(End)
LINKS
FORMULA
From Petros Hadjicostas, Jun 13 2019: (Start)
a(n) = Sum_{ceiling((n-1)/2) <= i <= n-1} G(i, n-1-i) for n >= 1, where G(n, k) = A140996(n, k) for 0 <= k <= n.
G.f.: x*(1 - x^2 - x^3 - x^4 - x^5)/((1 - x)*(1 + x)*(1 - x - x^2 - x^3 - x^4 - x^5)) = x*(1 - x^2 - x^3 - x^4 - x^5)/(1 - x - 2*x^2 + x^6 + x^7).
Recurrence: a(n) = -(3 + (-1)^n)/2 + a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5) for n >= 6 with a(1) = a(2) = 1, a(3) = 2, a(4) = 3, and a(5) = 6.
(End)
EXAMPLE
Pascal-like triangle with y = 3 and z = 0 (i.e, A140996) begins as follows:
1, so a(1) = 1.
1 1, so a(2) = 1.
1 2 1, so a(3) = 1 + 1 = 2.
1 4 2 1, so a(4) = 1 + 2 = 3.
1 8 4 2 1, so a(5) = 1 + 4 + 1 = 6.
1 16 8 4 2 1, so a(6) = 1 + 8 + 2 = 11.
1 31 17 8 4 2 1, so a(7) = 1 + 16 + 4 + 1 = 22.
1 60 35 17 8 4 2 1, so a(8) = 1 + 31 + 8 + 2 = 42.
1 116 72 35 17 8 4 2 1, so a(9) = 1 + 60 + 17 + 4 + 1 = 83.
1 224 148 72 35 17 8 4 2 1, so a(10) = 1 + 116 + 35 + 8 + 2 = 162.
1 432 303 149 72 35 17 8 4 2 1, so a(11) = 1 + 224 + 72 + 17 + 4 + 1 = 319.
... [edited by Petros Hadjicostas, Jun 13 2019]
CROSSREFS
KEYWORD
nonn
AUTHOR
Juri-Stepan Gerasimov, Jul 16 2008
EXTENSIONS
Partially edited by N. J. A. Sloane, Jul 18 2008
Name edited by and more terms from Petros Hadjicostas, Jun 13 2019
STATUS
approved