A339208 Triangle read by rows T(n, k) = Sum_{h>=0} Bernoulli(h)*binomial(n, h)*Stirling2(n-h, k)*k^h for n>=1 and 1<=k<=n.

1, 0, 1, 0, 0, 1, 0, -1, 0, 1, 0, 0, -5, 0, 1, 0, 3, 0, -15, 0, 1, 0, 0, 49, 0, -35, 0, 1, 0, -17, 0, 357, 0, -70, 0, 1, 0, 0, -809, 0, 1701, 0, -126, 0, 1, 0, 155, 0, -13175, 0, 6195, 0, -210, 0, 1, 0, 0, 20317, 0, -120395, 0, 18711, 0, -330, 0, 1

Offset

1,13

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

René Gy, Bernoulli-Stirling Numbers, INTEGERS 20 (2020), #A67. See Table 2 p. 12.

Formula

T(n, k) = Sum_{h>=0} Bernoulli(h)*binomial(n, h)*Stirling2(n-h, k)*k^h.

Example

Triangle begins
  1;
  0  1;
  0  0  1;
  0 -1  0   1;
  0  0 -5   0   1;
  0  3  0 -15   0  1;
  0  0 49   0 -35  0  1;
  ...

Mathematica

T[n_, k_] := Sum[BernoulliB[j] * Binomial[n, j] * StirlingS2[n - j, k] * k^j, {j, 0, n - k}]; Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 28 2020 *)

Prog

(PARI) T(n, k) = sum(h=0, n-k, bernfrac(h)*binomial(n, h)*stirling(n-h, k, 2)*k^h);
(Magma)
T:= func< n, k |(&+[Binomial(n, j)*Bernoulli(j)*StirlingSecond(n-j, k)*k^j: j in [0..n-k]]) >;
[T(n, k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jul 21 2022
(SageMath)
def A339208(n, k): return sum( binomial(n, j)*bernoulli(j)*stirling_number2(n-j, k)*k^j for j in (0..n-k) )
flatten([[A339208(n, k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Jul 21 2022

Crossrefs

Cf. A339207, A339209.

Cf. A027641/A027642 (Bernoulli), A007318 (binomial), A008277 (Stirling2).

Sequence in context: A225749 A227985 A071086 * A198105 A347683 A339209

Adjacent sequences: A339205 A339206 A339207 * A339209 A339210 A339211

Keyword

sign,tabl

Author

Michel Marcus, Nov 27 2020

Status

approved