OFFSET
0,4
COMMENTS
Comment from Lucas Larsen, Aug 20 2024: (Start)
The nonzero entries in the n-th row appear to be the nonzero coefficients (up to sign) in the following:
Let c be a fixed point in (0,oo) and f a smooth function such that f(c) = c and f(f'(x)) = x in a neighborhood of c. Then the n-th derivative of f evaluated at c can be written as a Laurent polynomial in c with the (descending) coefficients in question.
For instance:
f'(c) = c
f''(c) = c^(-1)
f'''(c) = -c^(-4)
f''''(c) = 3c^(-7) + c^(-8)
(End)
FORMULA
E.g.f. satisfies: A(x,y) = 1 + Series_Reversion[Integral A(-x*y,y) dx].
T(n,k) = [x^n*y^k] n!*A(x,y) for k=0..n(n-1)/2, n>=0.
Row sums equal A144005.
A067146(n) = Sum_{k=0..n(n-1)/2} (-1)^k*T(n,k).
This is a signed version of table A014621 because setting f((1+x)/y):=A(-x*y,y)/y for fixed y>0 implies f(f(x))*f'(x)=-1 and f(1/y)=1/y, as in the second formula of A014621. Therefore, the row sums form A014623 and the unsigned row sums form A014622. - Roland Miyamoto, Jun 03 2024
EXAMPLE
Triangle begins (without suppressing leading zeros):
1;
1;
0, 1;
0,0, 3, -1;
0,0,0, 15, -10, 3, -1;
0,0,0,0, 105, -105, 55, -30, 10, -3, 1;
0,0,0,0,0, 945, -1260, 910, -630, 350, -168, 76, -30, 10, -3, 1;
0,0,0,0,0,0, 10395, -17325, 15750, -12880, 9135, -5789, 3381, -1806, 910, -434, 196, -76, 30, -10, 3, -1;
0,0,0,0,0,0,0, 135135, -270270, 294525, -275275, 228375, -172200, 120960, -78519, 48006, -28336, 16065, -8609, 4461, -2166, 1018, -470, 196, -76, 30, -10, 3, -1; ...
PROG
(PARI) {T(n, k)=local(A=1+x*O(x^n)); for(i=0, n, A=1+serreverse(intformal(subst(A, x, -x*y)))); n!*polcoeff(polcoeff(A, n, x), k, y)}
(Python)
#This is only correct if the observation in the comment from 2024/08/20 is true.
def T(n, k):
if 0 <= n <= 1:
return 1 if k == 0 else 0
c = {(-1, ):1} #Polynomial in infinitely many variables (function iterates)
for _ in range(n-1):
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
_, row = zip(*sorted(pairs.items())) #Coefficients
if 0 <= k-n+1 < len(row): #Correcting number of leading 0s
return (-1)**(n+k+1)*abs(row[k-n+1]) #Correcting signs
else:
return 0
# Lucas Larsen, Aug 22 2024
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
Paul D. Hanna, Sep 10 2008
STATUS
approved