This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A377666 #26 Mar 31 2025 01:45:17
%S A377666 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,
%T A377666 -5,-9,117,205,-361,-61,1,1,-6,-11,236,497,-3362,-2763,0,1,1,-7,-13,
%U A377666 415,981,-15123,-22265,24611,1385,1
%N A377666 Array read by ascending antidiagonals: A(n, k) = Sum_{j = 0..k} binomial(k, j) * Euler(j, 0) *(2*n)^j.
%H A377666 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.)
%F A377666 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.
%F A377666 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).
%F A377666 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).
%e A377666 Array A(n, k) starts:
%e A377666 [0] 1, 1, 1, 1, 1, 1, 1, ... A000012
%e A377666 [1] 1, 0, -1, 0, 5, 0, -61, ... A122045
%e A377666 [2] 1, -1, -3, 11, 57, -361, -2763, ... A212435
%e A377666 [3] 1, -2, -5, 46, 205, -3362, -22265, ... A225147
%e A377666 [4] 1, -3, -7, 117, 497, -15123, -95767, ... A156201
%e A377666 [5] 1, -4, -9, 236, 981, -47524, -295029, ... A377665
%e A377666 [6] 1, -5, -11, 415, 1705, -120125, -737891, ...
%e A377666 [7] 1, -6, -13, 666, 2717, -262086, -1599793, ...
%p A377666 GHZeta := (k, n, m) -> m^(k+1)*Zeta(0, -k, 1/(m*n)):
%p A377666 A := (n, k) -> ifelse(n = 0, 1, n^k*(GHZeta(k, n, 4) - GHZeta(k, n, 2))):
%p A377666 for n from 0 to 7 do lprint(seq(A(n, k), k = 0..7)) od;
%p A377666 # Alternative:
%p A377666 P := proc(n, k) local j; 2*I*(1 + add(binomial(k, j)*polylog(-j, I)*n^j, j = 0..k)) end:
%p A377666 A := n -> Im(P(n, k)): seq(lprint(seq(A(n, k), k = 0..7)), n = 0..7);
%p A377666 # Computing the transpose using polynomials P from A363393.
%p A377666 P := n -> add(binomial(n + 1, j)*bernoulli(j, 1)*(4^j - 2^j)*x^(j-1), j = 0..n+1)/(n + 1):
%p A377666 Column := (k, n) -> subs(x = -n, P(k)):
%p A377666 for k from 0 to 6 do seq(Column(k, n), n = 0..9) od;
%p A377666 # According to the definition:
%p A377666 A := (n, k) -> local j; add(binomial(k, j)*euler(j, 0)*(2*n)^j, j = 0..k):
%p A377666 seq(lprint(seq(A(n, k), k = 0..6)), n = 0..7);
%t A377666 A[n_, k_] := n^k (4^(k+1) HurwitzZeta[-k, 1/(4n)] - 2^(k + 1) HurwitzZeta[-k, 1/(2n)]);
%o A377666 (SageMath)
%o A377666 from mpmath import *
%o A377666 mp.dps = 32; mp.pretty = True
%o A377666 def T(n, k):
%o A377666 p = 2*I*(1+sum(binomial(k, j)*polylog(-j, I)*n^j for j in range(k+1)))
%o A377666 return int(imag(p))
%o A377666 for n in range(8): print([T(n, k) for k in range(7)])
%Y A377666 Rows: A000012, A122045, A212435, A225147, A156201, A377665.
%Y A377666 Cf. A377663 (column 3), A377664 (main diagonal), A363393 (column polynomials).
%K A377666 sign,tabl
%O A377666 0,12
%A A377666 _Peter Luschny_, Nov 05 2024