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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) ) ). 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
FORMULA
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).
EXAMPLE
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
MAPLE
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
PROG
(PARI) T(n, k) = numerator(sum(m=1, n, 1/(2*m-1)^(k+1))*k!*(-2)^k*2)
CROSSREFS
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)).
Sequence in context: A099380 A252852 A351483 * A373564 A154909 A185348
KEYWORD
sign,frac,tabl
AUTHOR
Thomas Scheuerle, Apr 21 2024
STATUS
approved

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Last modified July 23 23:47 EDT 2024. Contains 374575 sequences. (Running on oeis4.)