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A375720 Irregular triangle, read by rows: Coefficients of the polynomials P_n, n>=2 such that the series f(x) = c + c(x-c) + Sum_{n>=2} P_n(c)/c^((n-1)*(n+2)/2+1) (x-c)^n/n! satisfies f(c) = c and f'(f(x)) = x near the fixed point c in (0,oo).

%I A375720 #17 Sep 12 2024 21:38:27
%S A375720 1,-1,1,3,-1,-3,-10,-15,1,3,10,30,55,105,105,-1,-3,-10,-30,-76,-168,
%T A375720 -350,-630,-910,-1260,-945,1,3,10,30,76,196,434,910,1806,3381,5789,
%U A375720 9135,12880,15750,17325,10395,-1,-3,-10,-30,-76,-196,-470,-1018,-2166,-4461,-8609,-16065,-28336,-48006,-78519,-120960,-172200,-228375,-275275,-294525,-270270,-135135
%N A375720 Irregular triangle, read by rows: Coefficients of the polynomials P_n, n>=2 such that the series f(x) = c + c(x-c) + Sum_{n>=2} P_n(c)/c^((n-1)*(n+2)/2+1) (x-c)^n/n! satisfies f(c) = c and f'(f(x)) = x near the fixed point c in (0,oo).
%C A375720 The indices in each row range from 0 to (n-3)*(n-2)/2
%C A375720 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.
%e A375720 Triangle begins:
%e A375720   1;
%e A375720   -1;
%e A375720   1, 3;
%e A375720   -1, -3, -10, -15;
%e A375720   1, 3, 10, 30, 55, 105, 105;
%e A375720   -1, -3, -10, -30, -76, -168, -350, -630, -910, -1260, -945;
%e A375720   ...
%e A375720 Polynomials are:
%e A375720   P_2(c) = 1
%e A375720   P_3(c) = -1
%e A375720   P_4(c) = 1 + 3c
%e A375720   P_5(c) = -1 - 3c - 10c^2 - 15c^3
%e A375720   etc.
%e A375720 Hence the series begins
%e A375720 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 + ...
%o A375720 (Python)
%o A375720 def T(n,k):
%o A375720     c = {(-1,):1} #Polynomial in infinitely many variables (function iterates)
%o A375720     for _ in range(n-2):
%o A375720         cnext = {}
%o A375720         for key, value in c.items():
%o A375720             key += (0,)
%o A375720             for i, ni in enumerate(key):
%o A375720                 term = tuple(nj-2 if j==i else nj-1 if j<=i+1 else nj
%o A375720                              for j,nj in enumerate(key))
%o A375720                 cnext[term] = cnext.get(term,0) + value*ni
%o A375720                 if cnext[term] == 0:
%o A375720                     del cnext[term]
%o A375720         c = cnext
%o A375720     pairs = {} #Reduction to single variable (evaluation at fixpoint)
%o A375720     for key, value in c.items():
%o A375720         s = sum(key)
%o A375720         pairs[s] = pairs.get(s,0) + value
%o A375720     return pairs.get(1+k-(n-1)*(n+2)//2,0)
%Y A375720 Cf. A144006.
%K A375720 sign,tabf
%O A375720 2,4
%A A375720 _Lucas Larsen_, Aug 26 2024