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) ) ).

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

Author

Thomas Scheuerle, Apr 21 2024

Keywords

• sign
• frac
• tabl

Examples

			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
		

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)).

Programs

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

PARI

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

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).