OFFSET
0,5
COMMENTS
From Petros Hadjicostas, Jun 10 2019: (Start)
According to the attached picture, the index of asymmetry here is s = 1 and the index of obliqueness (or obliquity) is e = 0.
In the picture, the equation G(n, e*n) = 1 becomes G(n, 0) = 1, while the equations G(n+x+1, n-e*n+e*x-e+1) = 2^x for 0 <= x < s = 1 become G(n+1, n+1) = 1 and G(n+2, n+1) = 2.
Also, in the picture, the recurrence G(n+s+2, k) = G(n+1, k-e*s+e-1) + Sum_{m=1..s+1} G(n+m, k-e*s+m*e-2*e) for k = 1..n+1 becomes G(n+3, k) = G(n+1, k-1) + G(n+1, k) + G(n+2, k) for k = 1..n+1.
Except for a shifting of the indices by 1, this array is a mirror image of array A140993. We have G(n, k) = A140993(n+1, n-k+1) for 0 <= k <= n. Triangular array A140993 has the same index of asymmetry (i.e., s = 1) but index of obliqueness e = 1.
(End)
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flatten
Juri-Stepan Gerasimov, Stepan's triangles and Pascal's triangle are connected by the recurrence relation ...
FORMULA
From Petros Hadjicostas, Jun 10 2019: (Start)
G(n, k) = A140993(n+1, n-k+1) for 0 <= k <= n.
Let A(x,y) = Sum_{n,k >= 0} G(n, k)*x^n*y^k and B(x,y) = Sum_{n,k >= 1} A140993(n, k). Then A(x, y) = x^(-1) * B(x*y, y^(-1)). Thus, the g.f. of the current array is A(x, y) = (1 - x - x^2 + x^3*y)/((1 - x) * (1 - x*y) * (1 - x - x^2 - x^2*y)).
To find the g.f. of the k-th column (where k >= 0), we differentiate A(x, y) k times with respect to y, divide by k!, and substitute y = 0. For example, differentiating A(x, y) once w.r.t. y and setting y = 0, we get the g.f. of the k = 1 column: x/((1 - x)*(1 - x - x^2)). This is the g.f. of sequence (A000071(n+2): n >= 0) = (Fibonacci(n+2) - 1: n >= 0).
G.f. of column k = 2 is x^2*(1 - x + x^3)/((1 - x)*(1 - x - x^2)^2). Thus, column k = 2 is a shifted version of (A140992(n): n >= 0).
(End)
EXAMPLE
Triangle begins (with rows for n >= 0 and columns for k >= 0):
1;
1, 1;
1, 2, 1;
1, 4, 2, 1;
1, 7, 5, 2, 1;
1, 12, 11, 5, 2, 1;
1, 20, 23, 12, 5, 2, 1;
1, 33, 46, 28, 12, 5, 2, 1;
1, 54, 89, 63, 29, 12, 5, 2, 1;
1, 88, 168, 137, 69, 29, 12, 5, 2, 1;
1, 143, 311, 289, 161, 70, 29, 12, 5, 2, 1;
MATHEMATICA
G[n_, k_] := G[n, k] = Which[k==0 || k==n, 1, k==n-1, 2, True, G[n-2, k-1] + G[n-2, k] + G[n-1, k]]; Table[G[n, k], {n, 0, 12}, {k, 0, n}] (* Jean-François Alcover, Jun 09 2019 *)
PROG
(PARI) G(n, k) = if(k==0 || k==n, 1, if(k==n-1, 2, G(n-1, k) + G(n-2, k) + G(n-2, k-1)));
for(n=0, 12, for(k=0, n, print1(G(n, k), ", "))) \\ G. C. Greubel, Jun 09 2019
(Sage)
def G(n, k):
if (k==0 or k==n): return 1
elif (k==n-1): return 2
else: return G(n-1, k) + G(n-2, k) + G(n-2, k-1)
[[G(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jun 09 2019
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Juri-Stepan Gerasimov, Jul 08 2008
EXTENSIONS
Indices in the definition corrected by R. J. Mathar, Aug 02 2009
Name edited by Petros Hadjicostas, Jun 10 2019
STATUS
approved