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 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. 3
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 * A199916 A363041 A180494 Adjacent sequences: A339204 A339205 A339206 * A339208 A339209 A339210 KEYWORD sign,tabl AUTHOR Michel Marcus, Nov 27 2020 STATUS approved

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Last modified June 7 12:23 EDT 2023. Contains 363157 sequences. (Running on oeis4.)