A377666 Array read by ascending antidiagonals: A(n, k) = Sum_{j = 0..k} binomial(k, j) * Euler(j, 0) *(2*n)^j.

1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, -3, 0, 1, 1, -3, -5, 11, 5, 1, 1, -4, -7, 46, 57, 0, 1, 1, -5, -9, 117, 205, -361, -61, 1, 1, -6, -11, 236, 497, -3362, -2763, 0, 1, 1, -7, -13, 415, 981, -15123, -22265, 24611, 1385, 1

Offset: 0

Author

Peter Luschny, Nov 05 2024

Keywords

• sign
• tabl

Examples

			Array A(n, k) starts:
  [0]  1,  1,   1,   1,    1,       1,        1, ...  A000012
  [1]  1,  0,  -1,   0,    5,       0,      -61, ...  A122045
  [2]  1, -1,  -3,  11,   57,    -361,    -2763, ...  A212435
  [3]  1, -2,  -5,  46,  205,   -3362,   -22265, ...  A225147
  [4]  1, -3,  -7, 117,  497,  -15123,   -95767, ...  A156201
  [5]  1, -4,  -9, 236,  981,  -47524,  -295029, ...  A377665
  [6]  1, -5, -11, 415, 1705, -120125,  -737891, ...
  [7]  1, -6, -13, 666, 2717, -262086, -1599793, ...
		

Links

• Peter Luschny, Generalized Eulerian polynomials. (See last row of the table.)

Crossrefs

Rows: A000012, A122045, A212435, A225147, A156201, A377665.

Cf. A377663 (column 3), A377664 (main diagonal), A363393 (column polynomials).

Programs

Maple

GHZeta := (k, n, m) -> m^(k+1)*Zeta(0, -k, 1/(m*n)):
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(A(n, k), k = 0..7)) od;
# Alternative:
P := proc(n, k) local j; 2*I*(1 + add(binomial(k, j)*polylog(-j, I)*n^j, j = 0..k)) end:
A := n -> Im(P(n, k)): seq(lprint(seq(A(n, k), k = 0..7)), n = 0..7);
# Computing the transpose using polynomials P from A363393.
P := n -> add(binomial(n + 1, j)*bernoulli(j, 1)*(4^j - 2^j)*x^(j-1), j = 0..n+1)/(n + 1):
Column := (k, n) -> subs(x = -n, P(k)):
for k from 0 to 6 do seq(Column(k, n), n = 0..9) od;
# According to the definition:
A := (n, k) -> local j; add(binomial(k, j)*euler(j, 0)*(2*n)^j, j = 0..k):
seq(lprint(seq(A(n, k), k = 0..6)), n = 0..7);

Mathematica

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

SageMath

from mpmath import *
mp.dps = 32; mp.pretty = True
def T(n, k):
    p = 2*I*(1+sum(binomial(k, j)*polylog(-j, I)*n^j for j in range(k+1)))
    return int(imag(p))
for n in range(8): print([T(n, k) for k in range(7)])

Formula

A(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.

A(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).

A(n, k) = substitute(x = -n, P(k, x)) where P(n, x) = (1/(n + 1)) * Sum_{j=0..n+1} binomial(n + 1, j) * Bernoulli(j, 1) * (4^j - 2^j)*x^(j-1).