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 A370691 #24 Apr 22 2024 17:44:30
%S A370691 1,1,1,3,1,1,15,9,1,1,105,225,27,1,765765,405810405,91398648466125,
%T A370691 48049812916875,1033788065625,89339709375,3796875,729,1,1,1,315,11025,
%U A370691 3375,27,1,1,3465,99225,1157625,16875,81,1,1,45045,12006225,31255875,40516875,253125,243,1,1,45045,2029052025
%N A370691 Square array read by upward antidiagonals: T(n, k) = denominator( 2*k!*(-2)^k*Sum_{m=1..n}( 1/(2*m-1)^(k+1) ) ).
%F A370691 T(n, k) = denominator( polygamma(k, n + 1/2) - polygamma(k, 1/2) ).
%F A370691 T(n, k) = denominator( k!*(-1)^(k+1)*(zeta((k+1), 1/2 + n) - zeta((k+1), 1/2)) ), where zeta is the Hurwitz zeta function.
%F A370691 T(n, 0) = A025547(n).
%F A370691 T(n, 1) = A128492(n).
%F A370691 Conjectured: T(n, 2) = A128507(n).
%e A370691 array begins:
%e A370691 1, 1, 1, 1, 1, 1
%e A370691 1, 1, 1, 1, 1, 1
%e A370691 3, 9, 27, 27, 81, 243
%e A370691 15, 225, 3375, 16875, 253125, 759375
%e A370691 105, 11025, 1157625, 40516875, 4254271875, 89339709375
%e A370691 315, 99225, 31255875, 3281866875, 1033788065625, 65128648134375
%e A370691 3465, 12006225, 41601569625, 48049812916875, 166492601756971875, 115379373017581509375
%p A370691 A := (n, k) -> Psi(k, n + 1/2) - Psi(k, 1/2):
%p A370691 seq(lprint(seq(denom(A(n, k)), k = 0..4)), n=0..6);
%o A370691 (PARI) T(n, k) = denominator(sum(m=1, n, 1/(2*m-1)^(k+1))*k!*(-2)^k*2)
%Y A370691 Cf. A370692 (numerators),
%Y A370691 Cf. A025547 (first column), A128492 (second column).
%Y A370691 Cf. A128507.
%Y A370691 Cf. A255008 (denominators polygamma(n, 1) - polygamma(n, k)).
%Y A370691 Cf. A255009 (numerators polygamma(n, 1) - polygamma(n, k)).
%K A370691 nonn,frac,tabl
%O A370691 0,4
%A A370691 _Thomas Scheuerle_, Apr 21 2024