A014621 - OEIS

%I #89 Feb 16 2024 20:49:58

%S 1,1,3,1,15,10,3,1,105,105,55,30,10,3,1,945,1260,910,630,350,168,76,

%T 30,10,3,1,10395,17325,15750,12880,9135,5789,3381,1806,910,434,196,76,

%U 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

%N Triangle of numbers arising from analysis of Levine's sequence A011784.

%H Roland Miyamoto, <a href="/A014621/b014621.txt">Rows n = 1..50 of triangle, flattened</a> (rows n = 1..6 from _Colin Mallows_)

%H Roland Miyamoto, <a href="/A014621/a014621_1.txt">Comments on A014621</a>, Oct 14 2022.

%H Roland Miyamoto and J. W. Sander, <a href="https://doi.org/10.1007/978-3-031-31617-3_14">Solving the iterative differential equation -gamma*g' = g^{-1}</a>, in: H. Maier, J. & R. Steuding (eds.), <a href="https://doi.org/10.1007/978-3-031-31617-3">Number Theory in Memory of Eduard Wirsing</a>, Springer, 2023, pp. 223-236, <a href="https://link.springer.com/chapter/10.1007/978-3-031-31617-3_14">alternative link</a>.

%H Roland Miyamoto, <a href="https://timebrain.org/Levine/a014621.py">Python3 program a014621.py</a> implementing below formulae.

%H Roland Miyamoto, <a href="https://arxiv.org/abs/2402.06618">Polynomial parametrisation of the canonical iterates to the solution of -gamma*g' = g^(-1)</a>, arXiv:2402.06618 [math.CO], 2024.

%F From _Roland Miyamoto_, Nov 20 2022: (Start)

%F 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.

%F 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   f^{(n)}(tau) = Sum_{k=0..(n-1)*(n-2)/2} (-1)^(n+k)*a(n,k)*tau^(2-3*n-k).

%F Thus, a(n,k) can be calculated recursively using the equations

%F   0 = (f ° f * f')^{(n)} = Sum_{k=0..n} binomial(n,k) (f ° f)^{(n-k)}*f^{(k+1)} for n=1,2,... (End)

%e Triangle begins:

%e     1;

%e     1;

%e     3,    1;

%e    15,   10,   3,   1;

%e   105,  105,  55,  30,  10,   3,  1;

%e   945, 1260, 910, 630, 350, 168, 76, 30, 10, 3, 1;

%e 10395, 17325, 15750, 12880, 9135, 5789, 3381, 1806, 910, 434, 196, 76, 30,

%e   10, 3, 1;

%e 135135, 270270, 294525, 275275, 228375, 172200, 120960, 78519, 48006, 28336, 16065, 8609, 4461, 2166, 1018, 470, 196, 76, 30, 10, 3, 1;

%e 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;

%o (Python) # See Miyamoto link.

%Y Cf. A011784, A014622 (row sums), A144006.

%K tabf,nonn

%O 1,3

%A _Colin Mallows_

%E More terms from _Roland Miyamoto_, Nov 20 2022

%E Offset corrected by _Max Alekseyev_, Sep 19 2023