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

0,5

Links

Table of n, a(n) for n=0..54.

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).

Example

[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;

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 *)

Crossrefs

Cf. A027641, A131689, A141056, A325871.

Sequence in context: A107681 A021298 A170952 * A175646 A324362 A073234

Adjacent sequences: A194584 A194585 A194586 * A194588 A194589 A194590

Keyword

sign,tabl

Author

Peter Luschny, Sep 17 2011

Extensions

Edited by Peter Luschny, Jun 26 2019

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

Status

approved