A255009
Array T(n,k) read by ascending antidiagonals, where T(n,k) is the denominator of polygamma(n, 1) - polygamma(n, k).
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 1, 1, 4, 6, 1, 1, 4, 36, 12, 1, 1, 8, 108, 144, 60, 1, 1, 4, 216, 864, 3600, 20, 1, 1, 8, 324, 3456, 108000, 3600, 140, 1, 1, 8, 1944, 10368, 2160000, 12000, 176400, 280, 1, 1, 16, 1944, 124416, 32400000, 2160000, 4116000
Offset: 0
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T[n_, k_] := (-1)^(n+1)*n!*HarmonicNumber[k-1, n+1] // Denominator; Table[T[n-k, k], {n, 0, 10}, {k, 1, n}] // Flatten
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) ) ).
Original entry on oeis.org
1, 1, 1, 3, 1, 1, 15, 9, 1, 1, 105, 225, 27, 1, 765765, 405810405, 91398648466125, 48049812916875, 1033788065625, 89339709375, 3796875, 729, 1, 1, 1, 315, 11025, 3375, 27, 1, 1, 3465, 99225, 1157625, 16875, 81, 1, 1, 45045, 12006225, 31255875, 40516875, 253125, 243, 1, 1, 45045, 2029052025
Offset: 0
array begins:
1, 1, 1, 1, 1, 1
1, 1, 1, 1, 1, 1
3, 9, 27, 27, 81, 243
15, 225, 3375, 16875, 253125, 759375
105, 11025, 1157625, 40516875, 4254271875, 89339709375
315, 99225, 31255875, 3281866875, 1033788065625, 65128648134375
3465, 12006225, 41601569625, 48049812916875, 166492601756971875, 115379373017581509375
Cf.
A255008 (denominators polygamma(n, 1) - polygamma(n, k)).
Cf.
A255009 (numerators polygamma(n, 1) - polygamma(n, k)).
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A := (n, k) -> Psi(k, n + 1/2) - Psi(k, 1/2):
seq(lprint(seq(denom(A(n, k)), k = 0..4)), n=0..6);
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T(n, k) = denominator(sum(m=1, n, 1/(2*m-1)^(k+1))*k!*(-2)^k*2)
A370692
Square array read by upward antidiagonals: T(n, k) = numerator( 2*k!*(-2)^k*Sum_{m=1..n}( 1/(2*m-1)^(k+1) ) ).
Original entry on oeis.org
0, 2, 0, 8, -4, 0, 46, -40, 16, 0, 352, -1036, 448, -96, 0, 1126, -51664, 56432, -2624, 768, 0, 13016, -469876, 19410176, -1642592, 62464, -7680, 0, 176138, -57251896, 524760752, -3945483392, 195262208, -1868800, 92160, 0, 176138, -57251896, 524760752, -3945483392, 195262208, -1868800, 92160
Offset: 0
array begins:
0, 0, 0, 0, 0
2, -4, 16, -96, 768
8, -40, 448, -2624, 62464
46, -1036, 56432, -1642592, 195262208
352, -51664, 19410176, -3945483392, 3281966329856
1126, -469876, 524760752, -319632174752, 797531263755008
13016, -57251896, 698956654912, -4680049729764032, 128444001508242193408
Cf.
A255008 (denominators polygamma(n, 1) - polygamma(n, k)).
Cf.
A255009 (numerators polygamma(n, 1) - polygamma(n, k)).
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A := (n, k) -> Psi(k, n + 1/2) - Psi(k, 1/2):
seq(lprint(seq(numer(A(n, k)), k = 0..4)), n=0..6); # Peter Luschny, Apr 22 2024
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T(n, k) = numerator(sum(m=1, n, 1/(2*m-1)^(k+1))*k!*(-2)^k*2)
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