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 A117396 #18 Aug 29 2024 18:54:53
%S A117396 1,1,1,1,2,1,1,5,3,1,1,17,11,4,1,1,77,51,19,5,1,1,437,291,109,29,6,1,
%T A117396 1,2957,1971,739,197,41,7,1,1,23117,15411,5779,1541,321,55,8,1,1,
%U A117396 204557,136371,51139,13637,2841,487,71,9,1,1,2018957,1345971,504739,134597,28041,4807,701,89,10,1
%N A117396 Triangle, read by rows, defined by: T(n,k) = (k+1)*T(n,k+1) - Sum_{j=1..n-k-1} T(j,0)*T(n,j+k+1) for n>k with T(n,n)=1 for n>=0.
%C A117396 Columns equal the partial sums of columns of triangle A092582 for k>0: T(n, k) - T(n-1, k) = A092582(n,k) = number of permutations p of [n] having length of first run equal to k.
%H A117396 G. C. Greubel, <a href="/A117396/b117396.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A117396 T(n,k) = k*Sum_{j=k-1..n} j!/(k+1)! for n >= k > 0, with T(n,0) = 1 for n >= 0. - _Paul D. Hanna_, Jun 20 2006
%e A117396 Triangle begins:
%e A117396 1;
%e A117396 1, 1;
%e A117396 1, 2, 1;
%e A117396 1, 5, 3, 1;
%e A117396 1, 17, 11, 4, 1;
%e A117396 1, 77, 51, 19, 5, 1;
%e A117396 1, 437, 291, 109, 29, 6, 1;
%e A117396 1, 2957, 1971, 739, 197, 41, 7, 1;
%e A117396 1, 23117, 15411, 5779, 1541, 321, 55, 8, 1;
%e A117396 1, 204557, 136371, 51139, 13637, 2841, 487, 71, 9, 1; ...
%e A117396 Matrix inverse is:
%e A117396 1;
%e A117396 -1, 1;
%e A117396 1, -2, 1;
%e A117396 1, 1, -3, 1;
%e A117396 1, 1, 1, -4, 1;
%e A117396 1, 1, 1, 1, -5, 1; ...
%e A117396 Matrix log is the integer triangle A117398:
%e A117396 0;
%e A117396 1, 0;
%e A117396 0, 2, 0;
%e A117396 -1, 2, 3, 0;
%e A117396 -3, 4, 5, 4, 0;
%e A117396 -9, 14, 15, 9, 5, 0;
%e A117396 -33, 68, 65, 34, 14, 6, 0; ...
%t A117396 T[n_, k_]:= T[n, k]= If[k==0, 1, k*Sum[j!/(k+1)!, {j,k-1,n}]];
%t A117396 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Sep 24 2021 *)
%o A117396 (PARI) T(n,k)=if(n<k || k<0,0,if(n==k,1,(k+1)*T(n,k+1)-sum(j=1,n-k-1,T(j,0)*T(n,j+k+1))))
%o A117396 (PARI) /* Definition by Matrix Inverse: */ T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,if(r==c+1,-c,1))));(M^-1)[n+1,k+1]
%o A117396 (PARI) T(n,k)=if(n<k || k<0,0,if(k==0,1,k*sum(j=k-1,n,j!)/(k+1)!)) \\ _Paul D. Hanna_, Jun 20 2006
%o A117396 (Magma) [k eq 0 select 1 else k*(&+[Factorial(j)/Factorial(k+1): j in [k-1..n]]): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 24 2021
%o A117396 (Sage)
%o A117396 def A117396(n,k): return 1 if (k==0) else k*sum(factorial(j)/factorial(k+1) for j in (k-1..n))
%o A117396 flatten([[A117396(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Sep 24 2021
%Y A117396 Cf. A014288 (column 1), A056199 (column 2), A117397 (column 3), A003422 (row sums), A117398 (matrix log); A092582.
%K A117396 nonn,tabl
%O A117396 0,5
%A A117396 _Paul D. Hanna_, Mar 11 2006