OFFSET
2,4
COMMENTS
The indices in each row range from 0 to (n-3)*(n-2)/2
When c = phi = (1+sqrt(5))/2 the series becomes the Taylor expansion of f(x) = phi^(-1/phi)*x^phi centered at phi, in particular the radius of convergence is positive for at least this choice of c.
EXAMPLE
Triangle begins:
1;
-1;
1, 3;
-1, -3, -10, -15;
1, 3, 10, 30, 55, 105, 105;
-1, -3, -10, -30, -76, -168, -350, -630, -910, -1260, -945;
...
Polynomials are:
P_2(c) = 1
P_3(c) = -1
P_4(c) = 1 + 3c
P_5(c) = -1 - 3c - 10c^2 - 15c^3
etc.
Hence the series begins
f(x) = c + c*(x-c) + c^(-1)(x-c)^2/2 - c^(-4)(x-c)^3/6 + (3c^(-7) + c^(-8))(x-c)^4/24 + ...
PROG
(Python)
def T(n, k):
c = {(-1, ):1} #Polynomial in infinitely many variables (function iterates)
for _ in range(n-2):
cnext = {}
for key, value in c.items():
key += (0, )
for i, ni in enumerate(key):
term = tuple(nj-2 if j==i else nj-1 if j<=i+1 else nj
for j, nj in enumerate(key))
cnext[term] = cnext.get(term, 0) + value*ni
if cnext[term] == 0:
del cnext[term]
c = cnext
pairs = {} #Reduction to single variable (evaluation at fixpoint)
for key, value in c.items():
s = sum(key)
pairs[s] = pairs.get(s, 0) + value
return pairs.get(1+k-(n-1)*(n+2)//2, 0)
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
Lucas Larsen, Aug 26 2024
STATUS
approved