OFFSET
2,1
COMMENTS
An interval is linear of height k if it is isomorphic to the total order on k+1 elements.
FORMULA
Row sums give A394191.
EXAMPLE
Triangle begins:
8, 8, 6, 4;
48, 72, 68, 52, 20, 8;
384, 768, 800, 608, 340, 200, 112, 48;
PROG
(SageMath)
def T(n, k):
if k < 0 or k > 2*n-1:
return 0
nW = factorial(n) * 2^n
if k == 0:
return nW
if k == 1:
return nW * n/2
coef = lambda h, i: sum(binomial(h+i+1, h-j) * binomial(j+i, j)
for j in range(h+1)) / 2^(h+1) / (h+i+1) / binomial(h+i, i)
s = 1/(k+1) if k%2 == 1 else 0
if k < n:
s += 2*(n-k)/(k+1)
if k == 3:
s += 1/4
s += 2 * sum(coef(k-2*i-1, i) for i in range(max(0, k-n), k//2))
return s * nW
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Ludovic Schwob, Mar 12 2026
STATUS
approved