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 A352650 #17 Apr 15 2022 13:02:11
%S A352650 1,0,1,1,1,1,2,4,2,1,9,15,9,3,1,44,76,44,16,4,1,265,455,265,95,25,5,1,
%T A352650 1854,3186,1854,666,174,36,6,1,14833,25487,14833,5327,1393,287,49,7,1,
%U A352650 133496,229384,133496,47944,12536,2584,440,64,8,1,1334961,2293839,1334961,479439,125361,25839,4401,639,81,9,1
%N A352650 Triangle read by rows: T(n,k) = n * T(n-1,k) + (-1)^(n-k) for 0 <= k <= n with initial values T(n,k) = 0 if n < 0 or k < 0 or k > n.
%C A352650 Conjecture 1: T(n,k) = Sum_{i=0..n-k} (-1)^(n+k+i) * A326326(n-k,i) * n^i for 0 <= k <= n.
%C A352650 Conjecture 2: T(n,k) = T(n-k,0) + Sum_{i=1..n-k} T(n-k,i) * T(i+k,k) * k / (i + k - 1) for 0 < k <= n.
%F A352650 T(n,n) = 1 for n >= 0.
%F A352650 T(n,n-1) = n - 1 for n > 0.
%F A352650 T(n,n-2) = (n - 1)^2 for n > 1.
%F A352650 T(n,0) = A000166(n) for n >= 0.
%F A352650 T(n,1) = A002467(n) for n > 0.
%F A352650 T(n,2) = A000166(n) for n > 1.
%F A352650 T(n,k) + T(n,k+1) = (n!) / (k!) for 0 <= k <= n.
%F A352650 T(n,k) = (n - 1) * (T(n-1,k) + T(n-2,k)) for 0 <= k < n-1.
%F A352650 T(n,k) = (T(n,k-2) - (k - 2) * T(n,k-1)) / (k - 1) for 1 < k <= n.
%F A352650 The row polynomials p(n,x) = Sum_{k=0..n} T(n,k) * x^k satisfy recurrence equation p(n,x) = (n - 1) * (p(n-1,x) + p(n-2,x)) + x^n for n > 0 with initial value p(0,x) = 1.
%F A352650 Row sums are p(n,1) = abs(A009179(n)) for n >= 0.
%F A352650 Alternating row sums are p(n,-1) = (-1)^n for n >= 0.
%F A352650 T(n,k) * T(n+1,k+1) - T(n+1,k) * T(n,k+1) = (-1)^(n-k) * A094587(n,k) for 0 <= k <= n.
%F A352650 Define 3x3-matrices T(i,j) with n <= i <= n+2 and k <= j <= k+2. Then we have: det(T(i,j)) = 0^(n-k) for 0 <= k <= n.
%F A352650 E.g.f. of column k >= 0: Sum_{n>=k} T(n,k) * t^n / (n!) = (Sum_{n>=k} (-t)^n / (n!)) * (-1)^k / (1 - t).
%F A352650 E.g.f.: Sum_{n>=0, k=0..n} T(n,k) * x^k * t^n / (n!) = (x * exp(x * t) + exp(-t)) / ((1 + x) * (1 - t)).
%F A352650 p(n,x) = Sum_{k=0..n} ((n!)/(k!))*(x^(k+1) + (-1)^k)/(x + 1) for n >= 0.
%F A352650 T(n,k) = Sum_{i=0..n-k} (-1)^i * (n!) / ((k+i)!) for 0 <= k <= n.
%F A352650 T(n,k) equals matrix product of A094587 and A097807.
%e A352650 The triangle T(n,k) for 0 <= k <= n starts:
%e A352650 n\k : 0 1 2 3 4 5 6 7 8 9
%e A352650 ================================================================
%e A352650 0 : 1
%e A352650 1 : 0 1
%e A352650 2 : 1 1 1
%e A352650 3 : 2 4 2 1
%e A352650 4 : 9 15 9 3 1
%e A352650 5 : 44 76 44 16 4 1
%e A352650 6 : 265 455 265 95 25 5 1
%e A352650 7 : 1854 3186 1854 666 174 36 6 1
%e A352650 8 : 14833 25487 14833 5327 1393 287 49 7 1
%e A352650 9 : 133496 229384 133496 47944 12536 2584 440 64 8 1
%e A352650 etc.
%p A352650 T := proc(n,k) option remember;
%p A352650 if k > n then 0 else n * T(n-1,k) + (-1)^(n-k) fi end:
%p A352650 for n from 0 to 9 do seq(T(n, k), k = 0..n) od; # _Peter Luschny_, Apr 11 2022
%Y A352650 Cf. A000166 (column 0 and 2), A002467 (column 1), A006347 (column 3), A006348 (column 4), A009179 (row sums, signed), A352988 (matrix inverse).
%Y A352650 Cf. A094587, A097807, A326326.
%K A352650 nonn,easy,tabl
%O A352650 0,7
%A A352650 _Werner Schulte_, Apr 04 2022