This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A144006 #28 Sep 12 2024 21:36:57
%S A144006 1,1,1,3,-1,15,-10,3,-1,105,-105,55,-30,10,-3,1,945,-1260,910,-630,
%T A144006 350,-168,76,-30,10,-3,1,10395,-17325,15750,-12880,9135,-5789,3381,
%U A144006 -1806,910,-434,196,-76,30,-10,3,-1,135135,-270270,294525,-275275,228375
%N 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.
%C A144006 Comment from _Lucas Larsen_, Aug 20 2024: (Start)
%C A144006 The nonzero entries in the n-th row appear to be the nonzero coefficients (up to sign) in the following:
%C A144006 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.
%C A144006 For instance:
%C A144006 f'(c) = c
%C A144006 f''(c) = c^(-1)
%C A144006 f'''(c) = -c^(-4)
%C A144006 f''''(c) = 3c^(-7) + c^(-8)
%C A144006 (End)
%F A144006 E.g.f. satisfies: A(x,y) = 1 + Series_Reversion[Integral A(-x*y,y) dx].
%F A144006 T(n,k) = [x^n*y^k] n!*A(x,y) for k=0..n(n-1)/2, n>=0.
%F A144006 Row sums equal A144005.
%F A144006 A067146(n) = Sum_{k=0..n(n-1)/2} (-1)^k*T(n,k).
%F A144006 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
%e A144006 Triangle begins (without suppressing leading zeros):
%e A144006 1;
%e A144006 1;
%e A144006 0, 1;
%e A144006 0,0, 3, -1;
%e A144006 0,0,0, 15, -10, 3, -1;
%e A144006 0,0,0,0, 105, -105, 55, -30, 10, -3, 1;
%e A144006 0,0,0,0,0, 945, -1260, 910, -630, 350, -168, 76, -30, 10, -3, 1;
%e A144006 0,0,0,0,0,0, 10395, -17325, 15750, -12880, 9135, -5789, 3381, -1806, 910, -434, 196, -76, 30, -10, 3, -1;
%e A144006 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; ...
%o A144006 (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)}
%o A144006 (Python)
%o A144006 #This is only correct if the observation in the comment from 2024/08/20 is true.
%o A144006 def T(n,k):
%o A144006 if 0 <= n <= 1:
%o A144006 return 1 if k == 0 else 0
%o A144006 c = {(-1,):1} #Polynomial in infinitely many variables (function iterates)
%o A144006 for _ in range(n-1):
%o A144006 cnext = {}
%o A144006 for key, value in c.items():
%o A144006 key += (0,)
%o A144006 for i, ni in enumerate(key):
%o A144006 term = tuple(nj-2 if j==i else nj-1 if j<=i+1 else nj
%o A144006 for j,nj in enumerate(key))
%o A144006 cnext[term] = cnext.get(term,0) + value*ni
%o A144006 if cnext[term] == 0:
%o A144006 del cnext[term]
%o A144006 c = cnext
%o A144006 pairs = {} #Reduction to single variable (evaluation at fixpoint)
%o A144006 for key, value in c.items():
%o A144006 s = -sum(key)
%o A144006 pairs[s] = pairs.get(s,0) + value
%o A144006 _, row = zip(*sorted(pairs.items())) #Coefficients
%o A144006 if 0 <= k-n+1 < len(row): #Correcting number of leading 0s
%o A144006 return (-1)**(n+k+1)*abs(row[k-n+1]) #Correcting signs
%o A144006 else:
%o A144006 return 0
%o A144006 # _Lucas Larsen_, Aug 22 2024
%Y A144006 Cf. A144005, A067146.
%Y A144006 Generates A014621, A014622 and A014623, which are related to Levine's sequence A011784.
%K A144006 sign,tabf
%O A144006 0,4
%A A144006 _Paul D. Hanna_, Sep 10 2008