A255008 Array T(n,k) read by ascending antidiagonals, where T(n,k) is the numerator of polygamma(n, 1) - polygamma(n, k).

0, 0, -1, 0, 1, -3, 0, -2, 5, -11, 0, 6, -9, 49, -25, 0, -24, 51, -251, 205, -137, 0, 120, -99, 1393, -2035, 5269, -49, 0, -720, 975, -8051, 22369, -256103, 5369, -363, 0, 5040, -5805, 237245, -257875, 14001361, -28567, 266681, -761, 0, -40320

Offset: 0

Author

Jean-François Alcover, Feb 12 2015

Keywords

• sign
• frac
• tabl
• easy

Comments

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

Examples

			Array of fractions begin:
0,  -1,  -3/2,       -11/6,          -25/12,               -137/60, ...
0,   1,   5/4,       49/36,         205/144,             5269/3600, ...
0,  -2,  -9/4,    -251/108,       -2035/864,        -256103/108000, ...
0,   6,  51/8,    1393/216,      22369/3456,      14001361/2160000, ...
0, -24, -99/4,   -8051/324,   -257875/10368,   -806108207/32400000, ...
0, 120, 975/8, 237245/1944, 15187325/124416, 47463376609/388800000, ...
...
		

Links

• Eric Weisstein's MathWorld, Harmonic Number.
• Eric Weisstein's MathWorld, Polygamma Function.
• Wikipedia, Polygamma Function.

Crossrefs

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

Programs

Mathematica

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

Formula

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.