A339208 - OEIS

%I #17 Jul 21 2022 01:55:19

%S 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,

%T 0,357,0,-70,0,1,0,0,-809,0,1701,0,-126,0,1,0,155,0,-13175,0,6195,0,

%U -210,0,1,0,0,20317,0,-120395,0,18711,0,-330,0,1

%N 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.

%H G. C. Greubel, <a href="/A339208/b339208.txt">Rows n = 1..50 of the triangle, flattened</a>

%H René Gy, <a href="http://math.colgate.edu/~integers/u67/u67.mail.html">Bernoulli-Stirling Numbers</a>, INTEGERS 20 (2020), #A67. See Table 2 p. 12.

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

%e Triangle begins

%e   1;

%e   0  1;

%e   0  0  1;

%e   0 -1  0   1;

%e   0  0 -5   0   1;

%e   0  3  0 -15   0  1;

%e   0  0 49   0 -35  0  1;

%e   ...

%t 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 *)

%o (PARI) T(n, k) = sum(h=0, n-k, bernfrac(h)*binomial(n,h)*stirling(n-h, k, 2)*k^h);

%o (Magma)

%o T:= func< n,k |(&+[Binomial(n,j)*Bernoulli(j)*StirlingSecond(n-j,k)*k^j: j in [0..n-k]]) >;

%o [T(n, k): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Jul 21 2022

%o (SageMath)

%o def A339208(n,k): return sum( binomial(n,j)*bernoulli(j)*stirling_number2(n-j,k)*k^j for j in (0..n-k) )

%o flatten([[A339208(n,k) for k in (1..n)] for n in (1..15)]) # _G. C. Greubel_, Jul 21 2022

%Y Cf. A339207, A339209.

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

%K sign,tabl

%O 1,13

%A _Michel Marcus_, Nov 27 2020