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 A370692 #17 Apr 22 2024 17:44:02
%S A370692 0,2,0,8,-4,0,46,-40,16,0,352,-1036,448,-96,0,1126,-51664,56432,-2624,
%T A370692 768,0,13016,-469876,19410176,-1642592,62464,-7680,0,176138,-57251896,
%U A370692 524760752,-3945483392,195262208,-1868800,92160,0,176138,-57251896,524760752,-3945483392,195262208,-1868800,92160
%N A370692 Square array read by upward antidiagonals: T(n, k) = numerator( 2*k!*(-2)^k*Sum_{m=1..n}( 1/(2*m-1)^(k+1) ) ).
%F A370692 T(n, k) = numerator( polygamma(k, n + 1/2) - polygamma(k, 1/2) ).
%F A370692 T(n, k) = numerator( k!*(-1)^(k+1)*(zeta((k+1), 1/2 + n) - zeta((k+1), 1/2)) ), where zeta is the Hurwitz zeta function.
%F A370692 T(n, 0) = A074599(n).
%F A370692 T(n, 1) = A173945(n+1).
%e A370692 array begins:
%e A370692 0, 0, 0, 0, 0
%e A370692 2, -4, 16, -96, 768
%e A370692 8, -40, 448, -2624, 62464
%e A370692 46, -1036, 56432, -1642592, 195262208
%e A370692 352, -51664, 19410176, -3945483392, 3281966329856
%e A370692 1126, -469876, 524760752, -319632174752, 797531263755008
%e A370692 13016, -57251896, 698956654912, -4680049729764032, 128444001508242193408
%p A370692 A := (n, k) -> Psi(k, n + 1/2) - Psi(k, 1/2):
%p A370692 seq(lprint(seq(numer(A(n, k)), k = 0..4)), n=0..6); # _Peter Luschny_, Apr 22 2024
%o A370692 (PARI) T(n, k) = numerator(sum(m=1, n, 1/(2*m-1)^(k+1))*k!*(-2)^k*2)
%Y A370692 Cf. A370691 (denominators).
%Y A370692 Cf. A074599 (first column), A173945 (second column).
%Y A370692 Cf. A255008 (denominators polygamma(n, 1) - polygamma(n, k)).
%Y A370692 Cf. A255009 (numerators polygamma(n, 1) - polygamma(n, k)).
%K A370692 sign,frac,tabl
%O A370692 0,2
%A A370692 _Thomas Scheuerle_, Apr 21 2024