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 A348482 #16 Jul 11 2024 17:12:42
%S A348482 1,2,1,4,3,1,10,9,4,1,34,33,16,5,1,154,153,76,25,6,1,874,873,436,145,
%T A348482 36,7,1,5914,5913,2956,985,246,49,8,1,46234,46233,23116,7705,1926,385,
%U A348482 64,9,1,409114,409113,204556,68185,17046,3409,568,81,10,1
%N A348482 Triangle read by rows: T(n,k) = (Sum_{i=k..n} i!)/(k!) for 0 <= k <= n.
%C A348482 The matrix inverse M = T^(-1) has terms M(n,n) = 1 for n >= 0, M(n,n-1) = -(n+1) for n > 0, and M(n,n-2) = n for n > 1, otherwise 0.
%H A348482 Sela Fried, <a href="/A348482/a348482.pdf">On a sum involving factorials</a>, 2024.
%F A348482 T(n,n) = 1 and T(2*n,n) = A109398(n) for n >= 0; T(n,n-1) = n+1 for n > 0; T(n,n-2) = n^2 for n > 1.
%F A348482 T(n,k) - T(n-1,k) = (n!) / (k!) = A094587(n,k) for 0 <= k < n.
%F A348482 T(n,k) = (k+2) * (T(n,k+1) - T(n,k+2)) for 0 <= k < n-1.
%F A348482 T(n,k) = (T(n,k-1) - 1) / k for 0 < k <= n.
%F A348482 T(n,k) * T(n-1,k-1) - T(n-1,k) * T(n,k-1) = (n!) / (k!) for 0 < k < n.
%F A348482 T(n,1) = T(n,0)-1 = Sum_{k=0..n-1} T(n,k)/(k+2) for n > 0 (conjectured).
%F A348482 Sum_{k=0..n} binomial(k+r,k) * (1-k) * T(n+r,k+r) = binomial(n+r+1,n) for n >= 0 and r >= 0.
%F A348482 Sum_{k=0..n} (-1)^k * (k+1) * T(n,k) = (1 + (-1)^n) / 2 for n >= 0.
%F A348482 Sum_{k=0..n} (-1)^k * (k!) * T(n,k) = Sum_{k=0..n} (k!) * (1+(-1)^k) / 2 for n >= 0.
%F A348482 The row polynomials p(n,x) = Sum_{k=0..n} T(n,k) * x^k for n >= 0 satisfy the following equations:
%F A348482 (a) p(n,x) - p'(n,x) = (x^(n+1)-1) / (x-1) for n >= 0, where p' is the first derivative of p;
%F A348482 (b) p(n,x) - (n+1) * p(n-1,x) + n * p(n-2,x) = x^n for n > 1.
%F A348482 (c) p(n,x) = (x+1) * p(n-1,x) + 1 + Sum_{i=1..n-1} (d/dx)^i p(n-1,x) for n > 0 (conjectured).
%F A348482 Row sums p(n,1) equal A002104(n+1) for n >= 0.
%F A348482 Alternating row sums p(n,-1) equal A173184(n) for n >= 0 (conjectured).
%F A348482 The three conjectures stated above are true. See links. - _Sela Fried_, Jul 11 2024.
%F A348482 From _Peter Luschny_, Jul 11 2024: (Start)
%F A348482 T(n, k) = (t(k) - t(n + 1)) / k!, where t(n) = (-1)^(n + 1) * Gamma(n + 1) * Subfactorial(-(n + 1)).
%F A348482 T(n, k) = A143122(n, k) / k!. (End)
%e A348482 The triangle T(n,k) for 0 <= k <= n starts:
%e A348482 n\k : 0 1 2 3 4 5 6 7 8 9
%e A348482 =================================================================
%e A348482 0 : 1
%e A348482 1 : 2 1
%e A348482 2 : 4 3 1
%e A348482 3 : 10 9 4 1
%e A348482 4 : 34 33 16 5 1
%e A348482 5 : 154 153 76 25 6 1
%e A348482 6 : 874 873 436 145 36 7 1
%e A348482 7 : 5914 5913 2956 985 246 49 8 1
%e A348482 8 : 46234 46233 23116 7705 1926 385 64 9 1
%e A348482 9 : 409114 409113 204556 68185 17046 3409 568 81 10 1
%e A348482 etc.
%t A348482 T[n_, k_] := Sum[i!, {i, k, n}]/k!; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Oct 20 2021 *)
%Y A348482 Cf. A109398, A094587, A002104 (row sums), A173184 (alt. row sums), A000012 (main diagonal), A000027(1st subdiagonal), A000290 (2nd subdiagonal), A081437 (3rd subdiagonal), A192398 (4th subdiagonal), A003422 (column 0), A007489 (column 1), A345889 (column 2), A143122.
%K A348482 nonn,easy,tabl
%O A348482 0,2
%A A348482 _Werner Schulte_, Oct 20 2021