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

1,3

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

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

Sequence in context: A130757 A181996 A144006 * A193966 A366120 A113378

Adjacent sequences: A014618 A014619 A014620 * A014622 A014623 A014624

Keyword

tabf,nonn

Author

Colin Mallows

Extensions

More terms from Roland Miyamoto, Nov 20 2022

Offset corrected by Max Alekseyev, Sep 19 2023

Status

approved