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A371898 Triangle read by rows: T(n, k) = n * k * (T(n-1, k-1) + T(n-1, k)) for k > 0 with initial values T(n, 0) = 1 and T(i, j) = 0 for j > i.

%I A371898 #9 Apr 14 2024 17:12:05
%S A371898 1,1,1,1,4,4,1,15,48,36,1,64,504,1008,576,1,325,5680,22680,31680,
%T A371898 14400,1,1956,72060,510480,1304640,1382400,518400,1,13699,1036224,
%U A371898 12233340,50823360,94046400,79833600,25401600,1,109600,16798768,318469536,2017814400,5794790400,8346240000,5893171200,1625702400
%N A371898 Triangle read by rows: T(n, k) = n * k * (T(n-1, k-1) + T(n-1, k)) for k > 0 with initial values T(n, 0) = 1 and T(i, j) = 0 for j > i.
%F A371898 T(n, k) = Sum_{i=k..n} A131689(i, k) * n! / (n-i)!.
%F A371898 T(n, k) = n! * k! * (Sum_{i=0..n-k} A048993(n-i, k) / i!).
%F A371898 T(n, k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k, i) * A320031(n, i).
%F A371898 Conjecture: E.g.f. of column k is exp(t) * t^k * k! / (Prod_{i=0..k} (1 - i*t)).
%F A371898 Conjecture: Sum_{k=0..n} (-1)^(n-k) * T(n, k) = A000166(n).
%F A371898 T(n, k) = A371766(n, k) * A371767(n, k). - _Peter Luschny_, Apr 14 2024
%e A371898 Lower triangular array starts:
%e A371898 n\k :  0      1        2         3         4         5         6         7
%e A371898 ==========================================================================
%e A371898   0 :  1
%e A371898   1 :  1      1
%e A371898   2 :  1      4        4
%e A371898   3 :  1     15       48        36
%e A371898   4 :  1     64      504      1008       576
%e A371898   5 :  1    325     5680     22680     31680     14400
%e A371898   6 :  1   1956    72060    510480   1304640   1382400    518400
%e A371898   7 :  1  13699  1036224  12233340  50823360  94046400  79833600  25401600
%e A371898   etc.
%t A371898 T[n_, k_] := Sum[(-1)^(k - j)*Binomial[k, j]*HypergeometricPFQ[{1, -n}, {}, -j], {j, 0, k}];
%t A371898 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten  (* _Peter Luschny_, Apr 12 2024 *)
%o A371898 (PARI) T(n, k) = if(k==0, 1, if(k > n, 0, n*k*(T(n-1, k-1) + T(n-1, k))))
%Y A371898 Cf. A000012 (column 0), A007526 (column 1), A001044 (main diagonal).
%Y A371898 Cf. A000166, A048993, A131689, A320031, A371766, A371767.
%K A371898 nonn,easy,tabl
%O A371898 0,5
%A A371898 _Werner Schulte_, Apr 11 2024