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 A340264 T(n, k) = Sum_{j=0..k} binomial(n, k - j)*Stirling2(n - k + j, j). Triangle read by rows, 0 <= k <= n. 5
 1, 0, 2, 0, 1, 4, 0, 1, 6, 8, 0, 1, 11, 24, 16, 0, 1, 20, 70, 80, 32, 0, 1, 37, 195, 340, 240, 64, 0, 1, 70, 539, 1330, 1400, 672, 128, 0, 1, 135, 1498, 5033, 7280, 5152, 1792, 256, 0, 1, 264, 4204, 18816, 35826, 34272, 17472, 4608, 512 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS A006905(n) = Sum_{k=0..n} A001035(k) * T(n, k). - Michael Somos, Jul 18 2021 T(n, k) is the number of idempotent relations R on [n] containing exactly k strongly connected components such that the following conditional statement holds for all x, y in [n]: If x, y are in distinct strongly connected components of R then (x, y) is not in R. - Geoffrey Critzer, Jan 10 2024 LINKS Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50) Štefan Schwarz, On idempotent binary relations on a finite set, Czechoslovak Mathematical Journal, Vol. 20 (1970), No. 4, 696-702. Eric Weisstein's World of Mathematics, Bell Polynomial. FORMULA T(n, k) = (-1)^n * n! * [t^k] [x^n] exp(t*(exp(-x) - x - 1)). n-th row polynomial R(n,x) = exp(-x)*Sum_{k >= 0} (x + k)^n * x^k/k! = Sum_{k = 0..n} binomial(n,k)*Bell(k,x)*x^(n-k), where Bell(n,x) denotes the n-th Bell polynomial. - Peter Bala, Jan 13 2022 EXAMPLE [0] 1; [1] 0, 2; [2] 0, 1, 4; [3] 0, 1, 6, 8; [4] 0, 1, 11, 24, 16; [5] 0, 1, 20, 70, 80, 32; [6] 0, 1, 37, 195, 340, 240, 64; [7] 0, 1, 70, 539, 1330, 1400, 672, 128; [8] 0, 1, 135, 1498, 5033, 7280, 5152, 1792, 256; [9] 0, 1, 264, 4204, 18816, 35826, 34272, 17472, 4608, 512; MAPLE egf := exp(t*(exp(-x) - x - 1)); ser := series(egf, x, 22): p := n -> coeff(ser, x, n); seq(seq((-1)^n*n!*coeff(p(n), t, k), k=0..n), n = 0..10); # Alternative: T := (n, k) -> add(binomial(n, k - j)*Stirling2(n - k + j, j), j=0..k): seq(seq(T(n, k), k = 0..n), n=0..9); # Peter Luschny, Feb 09 2021 MATHEMATICA T[ n_, k_] := Sum[ Binomial[n, k-j] StirlingS2[n-k+j, j], {j, 0 , k}]; (* Michael Somos, Jul 18 2021 *) PROG (PARI) T(n, k) = sum(j=0, k, binomial(n, j)*stirling(n-j, k-j, 2)); /* Michael Somos, Jul 18 2021 */ CROSSREFS Sum of row(n) is A000110(n+1). Sum of row(n) - 2^n is A058681(n). Alternating sum of row(n) is A109747(n). Cf. A001035, A006905, A137375, A186081, A121337. Sequence in context: A271466 A218581 A307177 * A291878 A131487 A230747 Adjacent sequences: A340261 A340262 A340263 * A340265 A340266 A340267 KEYWORD nonn,tabl AUTHOR Peter Luschny, Jan 08 2021 EXTENSIONS New name from Peter Luschny, Feb 09 2021 STATUS approved

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Last modified May 25 15:58 EDT 2024. Contains 372800 sequences. (Running on oeis4.)