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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.

%I A194587 #28 Aug 20 2022 07:31:32
%S A194587 1,0,1,0,-3,4,0,1,-4,3,0,-15,140,-270,144,0,1,-20,75,-96,40,0,-21,868,
%T A194587 -5670,13104,-12600,4320,0,1,-84,903,-3360,5600,-4320,1260,0,-15,2540,
%U A194587 -43470,244944,-630000,820800,-529200,134400,0,1,-340,9075,-74592,278040,-544320,582120,-322560,72576
%N 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.
%F A194587 T(n, k) = (-1)^(n - k) * A131689(n, k) * A141056(n) / (k + 1).
%F A194587 Sum_{k=0..n} T(n, k) = A164555(n).
%F A194587 T(n, n) = A325871(n).
%e A194587 [0] 1;
%e A194587 [1] 0,   1;
%e A194587 [2] 0,  -3,  4;
%e A194587 [3] 0,   1, -4,     3;
%e A194587 [4] 0, -15, 140, -270,    144;
%e A194587 [5] 0,   1, -20,   75,    -96,     40;
%e A194587 [6] 0, -21, 868, -5670, 13104, -12600,  4320;
%e A194587 [7] 0,   1, -84,   903, -3360,   5600, -4320, 1260;
%p A194587 A194587 := proc(n, k) local i;
%p A194587 mul(i, i = select(isprime, map(i -> i + 1, numtheory[divisors](n)))):
%p A194587 (-1)^(n-k)*Stirling2(n, k) * k! / (k + 1): %%*% end:
%p A194587 seq(print(seq(A194587(n, k), k = 0..n)), n = 0..7);
%t A194587 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 *)
%Y A194587 Cf. A027641, A131689, A141056, A325871.
%K A194587 sign,tabl
%O A194587 0,5
%A A194587 _Peter Luschny_, Sep 17 2011
%E A194587 Edited by _Peter Luschny_, Jun 26 2019
%E A194587 Edited and flipped signs in odd indexed rows by _Peter Luschny_, Aug 20 2022