A255009 -id:A255009 - cp's OEIS frontend

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

Jean-François Alcover, Feb 12 2015

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

			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, ...
...

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

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

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

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.

A370691 Square array read by upward antidiagonals: T(n, k) = denominator( 2k!(-2)^kSum_{m=1..n}( 1/(2m-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

Thomas Scheuerle, Apr 21 2024

			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. A370692 (numerators),
Cf. A025547 (first column), A128492 (second column).
Cf. A128507.
Cf. A255008 (denominators polygamma(n, 1) - polygamma(n, k)).
Cf. A255009 (numerators polygamma(n, 1) - polygamma(n, k)).

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);

T(n, k) = denominator(sum(m=1, n, 1/(2*m-1)^(k+1))*k!*(-2)^k*2)

T(n, k) = denominator( polygamma(k, n + 1/2) - polygamma(k, 1/2) ).
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.
T(n, 0) = A025547(n).
T(n, 1) = A128492(n).
Conjectured: T(n, 2) = A128507(n).

A370692 Square array read by upward antidiagonals: T(n, k) = numerator( 2k!(-2)^kSum_{m=1..n}( 1/(2m-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

Thomas Scheuerle, Apr 21 2024

			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. A370691 (denominators).
Cf. A074599 (first column), A173945 (second column).
Cf. A255008 (denominators polygamma(n, 1) - polygamma(n, k)).
Cf. A255009 (numerators polygamma(n, 1) - polygamma(n, k)).

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

T(n, k) = numerator(sum(m=1, n, 1/(2*m-1)^(k+1))*k!*(-2)^k*2)

T(n, k) = numerator( polygamma(k, n + 1/2) - polygamma(k, 1/2) ).
T(n, k) = numerator( k!*(-1)^(k+1)*(zeta((k+1), 1/2 + n) - zeta((k+1), 1/2)) ), where zeta is the Hurwitz zeta function.
T(n, 0) = A074599(n).
T(n, 1) = A173945(n+1).

cp's OEIS Frontend

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

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A370691 Square array read by upward antidiagonals: T(n, k) = denominator( 2k!(-2)^kSum_{m=1..n}( 1/(2m-1)^(k+1) ) ).

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A370692 Square array read by upward antidiagonals: T(n, k) = numerator( 2k!(-2)^kSum_{m=1..n}( 1/(2m-1)^(k+1) ) ).

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

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Crossrefs

Programs

Mathematica

Formula

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) ) ).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

PARI

Formula

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) ) ).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

PARI

Formula

A370691 Square array read by upward antidiagonals: T(n, k) = denominator( 2k!(-2)^kSum_{m=1..n}( 1/(2m-1)^(k+1) ) ).

A370692 Square array read by upward antidiagonals: T(n, k) = numerator( 2k!(-2)^kSum_{m=1..n}( 1/(2m-1)^(k+1) ) ).