A194587 A triangle whose rows add up to the numerators of the Bernoulli numbers (with B(1) = 1/2). T(n, k) for n >= 0, 0 <= k <= n.

1, 0, 1, 0, -3, 4, 0, 1, -4, 3, 0, -15, 140, -270, 144, 0, 1, -20, 75, -96, 40, 0, -21, 868, -5670, 13104, -12600, 4320, 0, 1, -84, 903, -3360, 5600, -4320, 1260, 0, -15, 2540, -43470, 244944, -630000, 820800, -529200, 134400, 0, 1, -340, 9075, -74592, 278040, -544320, 582120, -322560, 72576

Offset: 0

Author

Peter Luschny, Sep 17 2011

Keywords

• sign
• tabl

Examples

			[0] 1;
[1] 0,   1;
[2] 0,  -3,  4;
[3] 0,   1, -4,     3;
[4] 0, -15, 140, -270,    144;
[5] 0,   1, -20,   75,    -96,     40;
[6] 0, -21, 868, -5670, 13104, -12600,  4320;
[7] 0,   1, -84,   903, -3360,   5600, -4320, 1260;
		

Crossrefs

Cf. A027641, A131689, A141056, A325871.

Programs

Maple

A194587 := proc(n, k) local i;
mul(i, i = select(isprime, map(i -> i + 1, numtheory[divisors](n)))):
(-1)^(n-k)*Stirling2(n, k) * k! / (k + 1): %%*% end:
seq(print(seq(A194587(n, k), k = 0..n)), n = 0..7);

Mathematica

T[n_, k_] := Times @@ Select[Divisors[n]+1, PrimeQ] (-1)^(n-k) StirlingS2[n, k]* k!/(k+1); Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Jean-François Alcover, Jun 26 2019 *)

Formula

T(n, k) = (-1)^(n - k) * A131689(n, k) * A141056(n) / (k + 1).

Sum_{k=0..n} T(n, k) = A164555(n).

T(n, n) = A325871(n).

Extensions

Edited by Peter Luschny, Jun 26 2019

Edited and flipped signs in odd indexed rows by Peter Luschny, Aug 20 2022