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 A380179 #6 Feb 09 2025 14:58:25
%S A380179 1,1,1,1,5,1,1,14,14,1,1,33,68,30,1,1,72,257,218,55,1,1,151,873,1189,
%T A380179 553,91,1,1,310,2812,5734,4094,1204,140,1,1,629,8802,25916,26484,
%U A380179 11598,2352,204,1,1,1268,27107,112718,158840,96702,28566,4236,285,1
%N A380179 Triangle T(n,k) read by rows: T(n,k) = -binomial(n+1,k) + Sum_{i=0..k} Sum_{j=0..i+1} (i+1)^(n-i+j)*(-1)^(k-i)/(j!*(k-i)!) for 0 <= k <= n.
%F A380179 Conjecture: A347420(n) = 2^n + Sum_{k=1..n-1} T(n-1, k) for n >= 0.
%e A380179 Triangle begins:
%e A380179 1;
%e A380179 1, 1;
%e A380179 1, 5, 1;
%e A380179 1, 14, 14, 1;
%e A380179 1, 33, 68, 30, 1;
%e A380179 1, 72, 257, 218, 55, 1;
%e A380179 1, 151, 873, 1189, 553, 91, 1;
%e A380179 1, 310, 2812, 5734, 4094, 1204, 140, 1;
%e A380179 1, 629, 8802, 25916, 26484, 11598, 2352, 204, 1;
%e A380179 1, 1268, 27107, 112718, 158840, 96702, 28566, 4236, 285, 1;
%o A380179 (PARI) T(n,k) = if(k >= 0 && n >= k, -binomial(n+1, k) + sum(i=0, k, sum(j=0, i+1, (i+1)^(n-i+j)*(-1)^(k-i)/(j!*(k-i)!))))
%Y A380179 Cf. A347420.
%K A380179 nonn,tabl
%O A380179 0,5
%A A380179 _Mikhail Kurkov_, Jan 14 2025