This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A340264 #38 Jan 11 2024 01:18:02
%S A340264 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,
%T A340264 64,0,1,70,539,1330,1400,672,128,0,1,135,1498,5033,7280,5152,1792,256,
%U A340264 0,1,264,4204,18816,35826,34272,17472,4608,512
%N A340264 T(n, k) = Sum_{j=0..k} binomial(n, k - j)*Stirling2(n - k + j, j). Triangle read by rows, 0 <= k <= n.
%C A340264 A006905(n) = Sum_{k=0..n} A001035(k) * T(n, k). - _Michael Somos_, Jul 18 2021
%C A340264 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
%H A340264 Andrew Howroyd, <a href="/A340264/b340264.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H A340264 Štefan Schwarz, <a href="http://dx.doi.org/10.21136/CMJ.1970.100991">On idempotent binary relations on a finite set</a>, Czechoslovak Mathematical Journal, Vol. 20 (1970), No. 4, 696-702.
%H A340264 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F A340264 T(n, k) = (-1)^n * n! * [t^k] [x^n] exp(t*(exp(-x) - x - 1)).
%F A340264 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
%e A340264 [0] 1;
%e A340264 [1] 0, 2;
%e A340264 [2] 0, 1, 4;
%e A340264 [3] 0, 1, 6, 8;
%e A340264 [4] 0, 1, 11, 24, 16;
%e A340264 [5] 0, 1, 20, 70, 80, 32;
%e A340264 [6] 0, 1, 37, 195, 340, 240, 64;
%e A340264 [7] 0, 1, 70, 539, 1330, 1400, 672, 128;
%e A340264 [8] 0, 1, 135, 1498, 5033, 7280, 5152, 1792, 256;
%e A340264 [9] 0, 1, 264, 4204, 18816, 35826, 34272, 17472, 4608, 512;
%p A340264 egf := exp(t*(exp(-x) - x - 1));
%p A340264 ser := series(egf, x, 22):
%p A340264 p := n -> coeff(ser, x, n);
%p A340264 seq(seq((-1)^n*n!*coeff(p(n), t, k), k=0..n), n = 0..10);
%p A340264 # Alternative:
%p A340264 T := (n, k) -> add(binomial(n, k - j)*Stirling2(n - k + j, j), j=0..k):
%p A340264 seq(seq(T(n, k), k = 0..n), n=0..9); # _Peter Luschny_, Feb 09 2021
%t A340264 T[ n_, k_] := Sum[ Binomial[n, k-j] StirlingS2[n-k+j, j], {j, 0 ,k}]; (* _Michael Somos_, Jul 18 2021 *)
%o A340264 (PARI) T(n, k) = sum(j=0, k, binomial(n, j)*stirling(n-j, k-j, 2)); /* _Michael Somos_, Jul 18 2021 */
%Y A340264 Sum of row(n) is A000110(n+1).
%Y A340264 Sum of row(n) - 2^n is A058681(n).
%Y A340264 Alternating sum of row(n) is A109747(n).
%Y A340264 Cf. A001035, A006905, A137375, A186081, A121337.
%K A340264 nonn,tabl
%O A340264 0,3
%A A340264 _Peter Luschny_, Jan 08 2021
%E A340264 New name from _Peter Luschny_, Feb 09 2021