OFFSET
1,3
LINKS
Roland Miyamoto, Rows n = 1..50 of triangle, flattened (rows n = 1..6 from Colin Mallows)
Roland Miyamoto, Comments on A014621, Oct 14 2022.
Roland Miyamoto and J. W. Sander, Solving the iterative differential equation -gamma*g' = g^{-1}, in: H. Maier, J. & R. Steuding (eds.), Number Theory in Memory of Eduard Wirsing, Springer, 2023, pp. 223-236, alternative link.
Roland Miyamoto, Python3 program a014621.py implementing below formulae.
Roland Miyamoto, Polynomial parametrisation of the canonical iterates to the solution of -gamma*g' = g^(-1), arXiv:2402.06618 [math.CO], 2024.
FORMULA
From Roland Miyamoto, Nov 20 2022: (Start)
The n-th row contains 1 + (n-1)*(n-2)/2 numbers a(n,k), where n >= 1 and k = 0..(n-1)*(n-2)/2.
Let f be a solution to the iterative differential equation f(f(x))*f'(x) = -1 defined on some nonnegative interval and let tau=f(tau) be a fixed point of f. Then the n-th derivative of f at tau is
f^{(n)}(tau) = Sum_{k=0..(n-1)*(n-2)/2} (-1)^(n+k)*a(n,k)*tau^(2-3*n-k).
Thus, a(n,k) can be calculated recursively using the equations
0 = (f ° f * f')^{(n)} = Sum_{k=0..n} binomial(n,k) (f ° f)^{(n-k)}*f^{(k+1)} for n=1,2,... (End)
EXAMPLE
Triangle begins:
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, 172200, 120960, 78519, 48006, 28336, 16065, 8609, 4461, 2166, 1018, 470, 196, 76, 30, 10, 3, 1;
2027025, 4729725, 5990985, 6276270, 5853925, 4996530, 3999765, 2997225, 2115960, 1432725, 938644, 593646, 364551, 215940, 123639, 68886, 37276, 19485, 9959, 4911, 2301, 1063, 470, 196, 76, 30, 10, 3, 1;
PROG
(Python) # See Miyamoto link.
CROSSREFS
KEYWORD
tabf,nonn
AUTHOR
EXTENSIONS
More terms from Roland Miyamoto, Nov 20 2022
Offset corrected by Max Alekseyev, Sep 19 2023
STATUS
approved