A014621 Triangle of numbers arising from analysis of Levine's sequence A011784.
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
Offset: 1
Examples
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;
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.
Programs
-
Python
# See Miyamoto link.
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)
Extensions
More terms from Roland Miyamoto, Nov 20 2022
Offset corrected by Max Alekseyev, Sep 19 2023