A339207 Triangle read by rows T(n, k) = Sum_{h>=0} Bernoulli(h)*binomial(k+h-1, h)*abs(Stirling1(n, h+k))*n^h for n>=0 and 0<=k<=n.

1, 0, 1, 0, 0, 1, 0, -1, 0, 1, 0, 0, -5, 0, 1, 0, 24, 0, -15, 0, 1, 0, 0, 238, 0, -35, 0, 1, 0, -3396, 0, 1281, 0, -70, 0, 1, 0, 0, -51508, 0, 4977, 0, -126, 0, 1, 0, 1706112, 0, -408700, 0, 15645, 0, -210, 0, 1, 0, 0, 35028576, 0, -2267320, 0, 42273, 0, -330, 0, 1

Offset

0,13

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

René Gy, Bernoulli-Stirling Numbers, INTEGERS 20 (2020), #A67. See Table 1 p. 9.

Formula

T(n, k) = Sum_{h>=0} Bernoulli(h)*binomial(k+h-1, h)*abs(Stirling1(n, h+k))*n^h.

Example

Triangle begins
  1;
  0   1;
  0   0   1;
  0  -1   0   1;
  0   0  -5   0   1;
  0  24   0 -15   0  1;
  0   0 238   0 -35  0  1;
  ...

Mathematica

T[0, 0] = 1; T[n_, k_] := Sum[BernoulliB[j] * Binomial[k + j - 1, j] * Abs[StirlingS1[n, k + j]] * n^j, {j, 0, n - k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Nov 28 2020 *)

Prog

(PARI) T(n, k) = sum(h=0, n-k, bernfrac(h)*binomial(k+h-1, h)*abs(stirling(n, h+k, 1))*n^h);
(Magma)
function T(n, k)
      if k eq n then return 1;
    else return (&+[Binomial(k+j-1, j)*Bernoulli(j)*(-1)^j*StirlingFirst(n, k+j)*n^j: j in [0..n-k]]);
      end if; return T; end function;
[T(n, k): k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 21 2022
(SageMath)
def A339207(n, k):
    if (k==n): return 1
    else: return sum( binomial(k+j-1, j)*bernoulli(j)*stirling_number1(n, k+j)*n^j for j in (0..n-k) )
flatten([[A339207(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jul 21 2022

Crossrefs

Cf. A339208, A339209.

Cf. A027641/A027642 (Bernoulli), A007318 (binomial), A000254 (unsigned Stirling1).

Sequence in context: A132795 A277031 A085198 * A372959 A199916 A363041

Adjacent sequences: A339204 A339205 A339206 * A339208 A339209 A339210

Keyword

sign,tabl

Author

Michel Marcus, Nov 27 2020

Status

approved