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).
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, 1, 3, 10, 30, 76, 196, 434, 910, 1806, 3381, 5789, 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
Offset: 2
Examples
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 + ...
Crossrefs
Cf. A144006.
Programs
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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)
Comments