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 A342111 #23 May 14 2025 04:18:07
%S A342111 1,0,1,12,193,3980,100805,3034920,105994833,4215106728,188097696345,
%T A342111 9309515255700,506149663220641,29989851619249236,1923467938147053389,
%U A342111 132771455705186298000,9814431285244231295265,773520674985391641371280,64752473306596841023424945
%N A342111 a(n) = (-1)^n * Sum_{k=0..n} Stirling1(n,k) * Stirling1(n,n-k).
%H A342111 G. C. Greubel, <a href="/A342111/b342111.txt">Table of n, a(n) for n = 0..350</a>
%F A342111 a(n) ~ c * d^n * (n-1)!, where
%F A342111 d = A238261 = -(2*LambertW(-1,-exp(-1/2)/2))^2 / (1 + 2*LambertW(-1,-exp(-1/2)/2)) = 4.9108149645682558987515348052403521978987052817678471761394112...
%F A342111 c = 1/(4*sqrt(-LambertW(-1, -exp(-1/2)/2)) * sqrt(-1 - LambertW(-1, -exp(-1/2)/2))*Pi) = 0.06903826111269387517867145566264007373042059749428879149076344304196548... - _Vaclav Kotesovec_, Feb 28 2021, updated May 14 2025
%F A342111 a(n) = [x^n] Product_{k=0..n-1} (1 + k*x)^2. - _Seiichi Manyama_, May 13 2025
%t A342111 Table[(-1)^n*Sum[StirlingS1[n, k]*StirlingS1[n, n-k], {k, 0, n}], {n, 0, 20}]
%o A342111 (PARI) a(n) = (-1)^n*sum(k=0, n, stirling(n, k, 1)*stirling(n, n-k, 1)); \\ _Michel Marcus_, Feb 28 2021
%o A342111 (Magma) [(&+[(-1)^n*StirlingFirst(n, k)*StirlingFirst(n, n-k): k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Jun 03 2021
%o A342111 (Sage) [sum( stirling_number1(n, k)*stirling_number1(n, n-k) for k in (0..n) ) for n in (0..30)] # _G. C. Greubel_, Jun 03 2021
%Y A342111 Cf. A008275, A014322, A047796, A132393, A342110.
%Y A342111 Cf. A048994, A155826.
%Y A342111 Cf. A129256, A187235, A246117.
%K A342111 nonn
%O A342111 0,4
%A A342111 _Vaclav Kotesovec_, Feb 28 2021