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 A255008 #12 Feb 16 2025 08:33:24
%S A255008 0,0,-1,0,1,-3,0,-2,5,-11,0,6,-9,49,-25,0,-24,51,-251,205,-137,0,120,
%T A255008 -99,1393,-2035,5269,-49,0,-720,975,-8051,22369,-256103,5369,-363,0,
%U A255008 5040,-5805,237245,-257875,14001361,-28567,266681,-761,0,-40320
%N A255008 Array T(n,k) read by ascending antidiagonals, where T(n,k) is the numerator of polygamma(n, 1) - polygamma(n, k).
%C A255008 Up to signs, row n=0 is A001008/A002805, row n=1 is A007406/A007407 and column k=1 is n!.
%H A255008 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>.
%H A255008 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/PolygammaFunction.html">Polygamma Function</a>.
%H A255008 Wikipedia, <a href="http://en.wikipedia.org/wiki/Polygamma_function">Polygamma Function</a>.
%F A255008 Fraction giving T(n,k) = polygamma(n, 1) - polygamma(n, k) = (-1)^(n+1)*n! * sum_{j=1..k-1} 1/j^(n+1) = (-1)^(n+1)*n!*H(k-1, n+1), where H(n,r) gives the n-th harmonic number of order r.
%e A255008 Array of fractions begin:
%e A255008 0, -1, -3/2, -11/6, -25/12, -137/60, ...
%e A255008 0, 1, 5/4, 49/36, 205/144, 5269/3600, ...
%e A255008 0, -2, -9/4, -251/108, -2035/864, -256103/108000, ...
%e A255008 0, 6, 51/8, 1393/216, 22369/3456, 14001361/2160000, ...
%e A255008 0, -24, -99/4, -8051/324, -257875/10368, -806108207/32400000, ...
%e A255008 0, 120, 975/8, 237245/1944, 15187325/124416, 47463376609/388800000, ...
%e A255008 ...
%t A255008 T[n_, k_] := (-1)^(n+1)*n!*HarmonicNumber[k-1, n+1] // Numerator; Table[T[n-k, k], {n, 0, 10}, {k, 1, n}] // Flatten
%Y A255008 Cf. A001008, A002805, A007406, A007407, A255006, A255007, A255009 (denominators).
%K A255008 sign,frac,tabl,easy
%O A255008 0,6
%A A255008 _Jean-François Alcover_, Feb 12 2015