A144006 Triangle, read by rows of coefficients of x^n*y^k for k=0..n(n-1)/2 for n>=0, defined by e.g.f.: A(x,y) = 1 + Series_Reversion( Integral A(-x*y,y) dx ), with leading zeros in each row suppressed.

1, 1, 1, 3, -1, 15, -10, 3, -1, 105, -105, 55, -30, 10, -3, 1, 945, -1260, 910, -630, 350, -168, 76, -30, 10, -3, 1, 10395, -17325, 15750, -12880, 9135, -5789, 3381, -1806, 910, -434, 196, -76, 30, -10, 3, -1, 135135, -270270, 294525, -275275, 228375

Offset: 0

Author

Paul D. Hanna, Sep 10 2008

Keywords

• sign
• tabf

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)

Examples

			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; ...
		

Crossrefs

Cf. A144005, A067146.

Generates A014621, A014622 and A014623, which are related to Levine's sequence A011784.

Programs

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

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