Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #24 Apr 22 2024 17:44:30
%S 1,1,1,3,1,1,15,9,1,1,105,225,27,1,765765,405810405,91398648466125,
%T 48049812916875,1033788065625,89339709375,3796875,729,1,1,1,315,11025,
%U 3375,27,1,1,3465,99225,1157625,16875,81,1,1,45045,12006225,31255875,40516875,253125,243,1,1,45045,2029052025
%N 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 T(n, k) = denominator( polygamma(k, n + 1/2) - polygamma(k, 1/2) ).
%F 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 T(n, 0) = A025547(n).
%F T(n, 1) = A128492(n).
%F Conjectured: T(n, 2) = A128507(n).
%e array begins:
%e 1, 1, 1, 1, 1, 1
%e 1, 1, 1, 1, 1, 1
%e 3, 9, 27, 27, 81, 243
%e 15, 225, 3375, 16875, 253125, 759375
%e 105, 11025, 1157625, 40516875, 4254271875, 89339709375
%e 315, 99225, 31255875, 3281866875, 1033788065625, 65128648134375
%e 3465, 12006225, 41601569625, 48049812916875, 166492601756971875, 115379373017581509375
%p A := (n, k) -> Psi(k, n + 1/2) - Psi(k, 1/2):
%p seq(lprint(seq(denom(A(n, k)), k = 0..4)), n=0..6);
%o (PARI) T(n, k) = denominator(sum(m=1, n, 1/(2*m-1)^(k+1))*k!*(-2)^k*2)
%Y Cf. A370692 (numerators),
%Y Cf. A025547 (first column), A128492 (second column).
%Y Cf. A128507.
%Y Cf. A255008 (denominators polygamma(n, 1) - polygamma(n, k)).
%Y Cf. A255009 (numerators polygamma(n, 1) - polygamma(n, k)).
%K nonn,frac,tabl
%O 0,4
%A _Thomas Scheuerle_, Apr 21 2024