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proposed

a(n) is the n-th derivative of e^e^x divided by e at point x=0. - Joan Ludevid, Nov 05 2024

a(n) is the n-th derivative of e^e^x divided by e at point x=0. - Joan Ludevid, Nov 05 2024

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Array read by ascending antidiagonals: TA(n, k) = n^k*(GHZeta(k, n, 4) - GHZeta(k, n, 2))) where GHZeta(k, n, m) = m^(k+1)*HurwitzZeta(-k, 1/(m*n)) for n > 0, and T(0, k) = 1.

TA(n, k) = Im(P(n, k)) where P(n, k) = 2*i*(1 + Sum_{j=0..k} binomial(k, j)*polylog(-j, i)*n^j.

Array A(n, k) starts:

T A := (n, k) -> ifelse(n = 0, 1, n^k*(GHZeta(k, n, 4) - GHZeta(k, n, 2))):

for n from 0 to 7 do lprint(seq(TA(n, k), k = 0..7)) od;

P := proc(n, k) local j; 2*I*(1 + add(binomial(k, j)*polylog(-j, I)*n^j, j = 0..k)) end:

T A := n -> Im(P(n, k)): seq(lprint(seq(TA(n, k), k = 0..7)), n = 0..7);

TA[n_, k_] := n^k (4^(k+1) HurwitzZeta[-k, 1/(4n)] - 2^(k + 1) HurwitzZeta[-k, 1/(2n)]);

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def eytzinger(n):

def e(t, k=1, i=0):

if (k < len(t)):

i = e(t, k * 2, i)

t[k] = i

i += 1

i = e(t, k * 2 + 1, i)

return i

t = [0] * (n+1)

e(t)

return t[1:]

major_index = lambda p: sum(i+1 for i in range(len(p)-1) if p[i] > p[i+1])

a = lambda n: major_index(eytzinger(n)) + 1

print([a(n) for n in range(0, 59)]) # Darío Clavijo, Nov 04 2024

(Python)

print([a(n) for n in range(0, 5967)]) # Darío Clavijo, Nov 05 2024

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(SageMath)

from mpmath import *

mp.dps = 32; mp.pretty = True

def EulerianPoly(k, n):

if x == 1: return k^n*factorial(n)

p = 2*I*(1+sum(binomial(n, j)*polylog(-j, I)*k^j for j in range(n+1)))

return int(imag(p))

for n in range(8): print([EulerianPoly(n, k) for k in range(7)])

[0] 1, 1, 1, 1, 1, 1, 1, 1, ... A000012

[1] 1, 0, -1, 0, 5, 0, -61, 0, ... A122045

[2] 1, -1, -3, 11, 57, -361, -2763, 24611, ... A212435

[3] 1, -2, -5, 46, 205, -3362, -22265, 515086, ... A225147

[4] 1, -3, -7, 117, 497, -15123, -95767, 4116837, ... A156201

[5] 1, -4, -9, 236, 981, -47524, -295029, 20208716, ... A377667A377665

[6] 1, -5, -11, 415, 1705, -120125, -737891, 73544935, ...

[7] 1, -6, -13, 666, 2717, -262086, -1599793, 218380506, ...

Rows: A000012, A122045, A212435, A225147, A156201, A377667A377665.

T(n, k) = Im(P(n, k)) where P(n, k) = 2*i*(1 + Sum_{j=0..k} binomial(k, j)*polylog(-j, i)*n^j.

# Alternative:

P := proc(n, k) local j; 2*I*(1+add(binomial(k, j)*polylog(-j, I)*n^j, j = 0..k)) end:

T := n -> Im(P(n, k)): seq(lprint(seq(T(n, k), k = 0..7)), n = 0..7);

Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/EulerianPolynomialsGeneralized#Assorted_values_of_the_polynomials">Generalized Eulerian polynomials</a>. (See last row of the table.)

Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/EulerianPolynomialsGeneralized#Assorted_values_of_the_polynomials">Generalized Eulerian polynomials</a>. (See last row of the table.)

Peter Luschny, <a href="http://oeis.org/wiki/https://oeis.org/wiki/User:Peter_Luschny/EulerianPolynomialsGeneralized#Assorted_values_of_the_polynomials">Generalized Eulerian polynomials</a>. (See last row of the table.)