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 A325139 #7 Apr 15 2019 14:14:55
%S A325139 1,2,1,6,7,1,24,47,15,1,120,342,179,26,1,720,2754,2070,485,40,1,5040,
%T A325139 24552,24574,8175,1075,57,1,40320,241128,305956,134449,24885,2086,77,
%U A325139 1,362880,2592720,4028156,2231012,541849,63504,3682,100,1
%N A325139 Triangle T(n, k) = [t^n] Gamma(n + k + m + t)/Gamma(k + m + t) for m = 2 and 0 <= k <= n, read by rows.
%F A325139 T(n, k) = Sum_{j=0..n-k} binomial(j+k, k)*abs(Stirling1(n, j+k))*(k+2)^j.
%e A325139 0: 1;
%e A325139 1: 2, 1;
%e A325139 2: 6, 7, 1;
%e A325139 3: 24, 47, 15, 1;
%e A325139 4: 120, 342, 179, 26, 1;
%e A325139 5: 720, 2754, 2070, 485, 40, 1;
%e A325139 6: 5040, 24552, 24574, 8175, 1075, 57, 1;
%e A325139 7: 40320, 241128, 305956, 134449, 24885, 2086, 77, 1;
%e A325139 8: 362880, 2592720, 4028156, 2231012, 541849, 63504, 3682, 100, 1;
%e A325139 9: 3628800, 30334320, 56231712, 37972304, 11563650, 1768809, 142632, 6054, 126, 1;
%e A325139 A: A000142, A001711, A001717, A001723, ...
%p A325139 T := (n, k) -> add(binomial(j+k, k)*(k+2)^j*abs(Stirling1(n, j+k)), j=0..n-k):
%p A325139 seq(seq(T(n, k), k=0..n), n=0..8);
%p A325139 # Note that for n > 16 Maple fails (at least in some versions) to compute the
%p A325139 # terms properly. Inserting 'simplify' or numerical evaluation might help.
%p A325139 A325139Row := proc(n) local ogf, ser; ogf := (n, k) -> GAMMA(n+k+2+x)/GAMMA(k+2+x);
%p A325139 ser := (n, k) -> series(ogf(n,k), x, k+2); seq(coeff(ser(n,k), x, k), k=0..n) end:
%p A325139 seq(A325139Row(n), n=0..9);
%Y A325139 Row sums are A325140.
%Y A325139 Columns are: A000142, A001711, A001717, A001723.
%Y A325139 Family: A307419 (m=0), A325137 (m=1), this sequence (m=2).
%K A325139 nonn,tabl
%O A325139 0,2
%A A325139 _Peter Luschny_, Apr 15 2019