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Array T(n,k) read by ascending antidiagonals, where T(n,k) is the numerator of polygamma(n, 1) - polygamma(n, k).
3

%I #11 Jan 05 2024 12:29:24

%S 0,0,-1,0,1,-3,0,-2,5,-11,0,6,-9,49,-25,0,-24,51,-251,205,-137,0,120,

%T -99,1393,-2035,5269,-49,0,-720,975,-8051,22369,-256103,5369,-363,0,

%U 5040,-5805,237245,-257875,14001361,-28567,266681,-761,0,-40320

%N Array T(n,k) read by ascending antidiagonals, where T(n,k) is the numerator of polygamma(n, 1) - polygamma(n, k).

%C Up to signs, row n=0 is A001008/A002805, row n=1 is A007406/A007407 and column k=1 is n!.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/PolygammaFunction.html">Polygamma Function</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Polygamma_function">Polygamma Function</a>.

%F 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 Array of fractions begin:

%e 0, -1, -3/2, -11/6, -25/12, -137/60, ...

%e 0, 1, 5/4, 49/36, 205/144, 5269/3600, ...

%e 0, -2, -9/4, -251/108, -2035/864, -256103/108000, ...

%e 0, 6, 51/8, 1393/216, 22369/3456, 14001361/2160000, ...

%e 0, -24, -99/4, -8051/324, -257875/10368, -806108207/32400000, ...

%e 0, 120, 975/8, 237245/1944, 15187325/124416, 47463376609/388800000, ...

%e ...

%t 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 Cf. A001008, A002805, A007406, A007407, A255006, A255007, A255009 (denominators).

%K sign,frac,tabl,easy

%O 0,6

%A _Jean-François Alcover_, Feb 12 2015