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) ) ).

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

0,4

Links

Table of n, a(n) for n=0..47.

Formula

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).

Example

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

Maple

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);

Prog

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

Crossrefs

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)).

Sequence in context: A144269 A144270 A110112 * A326800 A176225 A173917

Adjacent sequences: A370688 A370689 A370690 * A370692 A370693 A370694

Keyword

nonn,frac,tabl

Author

Thomas Scheuerle, Apr 21 2024

Status

approved