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A350512 Triangle read by rows with T(n,0) = 1 for n >= 0 and T(n,k) = binomial(n-1,k-1)*(2*k*(n-k) + n)/k for 0 < k <= n.

%I A350512 #17 Jan 18 2022 21:46:14
%S A350512 1,1,1,1,4,1,1,7,7,1,1,10,18,10,1,1,13,34,34,13,1,1,16,55,80,55,16,1,
%T A350512 1,19,81,155,155,81,19,1,1,22,112,266,350,266,112,22,1,1,25,148,420,
%U A350512 686,686,420,148,25,1,1,28,189,624,1218,1512,1218,624,189,28,1
%N A350512 Triangle read by rows with T(n,0) = 1 for n >= 0 and T(n,k) = binomial(n-1,k-1)*(2*k*(n-k) + n)/k for 0 < k <= n.
%C A350512 Depending on some fixed integer m there is a family of number triangles T(m; n,k) for 0 <= k <= n with entries: T(m; n,0) = 1 for n >= 0 and T(m; n,k) = binomial(n-1,k-1)*(m*k*(n-k) + n)/k for 0 < k <= n.
%C A350512 Special cases: m=0 (A007318), m=1 (A103450), and m=2 (this triangle).
%C A350512 Further properties: T(m; n,n) = 1 for n >= 0; T(m; n,k) = T(m; n,n-k) for 0 <= k <= n; T(m; 2*n,n) = A000108(n)*A086270(m,n+1) for n >= 0 and m > 0.
%C A350512 T(m; n,k) = T(m; n-1,k) + T(m; n-1,k-1) + m*binomial(n-2,k-1) for 0 < k < n.
%C A350512 G.f. of column k: (1 + m*k*x) * x^k / (1 - x)^(k+1).
%C A350512 G.f.: A(x, t) = (1 - (1+x)*t + m*x*t^2) / (1 - (1+x)*t)^2.
%C A350512 T(m; n,k) = [x^k] (1 + (m*n - m + 2)*x + x^2) * (1 + x)^(n-2) for n > 0.
%F A350512 T(n, n) = 1; T(n, k) = T(n, n-k).
%F A350512 T(2*n, n) = (n+1)^2 * A000108(n).
%F A350512 T(n, k) = T(n-1, k) + T(n-1, k-1) + 2 * binomial(n-2,k-1) for 0 < k < n.
%F A350512 G.f. of column k: (1 + 2*k*x) * x^k / (1 - x)^(k+1).
%F A350512 G.f.: A(x,t) = (1 - (1 + x) * t + 2 * x * t^2) / (1 - (1 + x) * t)^2.
%F A350512 T(n,k) = [x^k] (1 + 2 * n * x + x^2) * (1 + x)^(n-2) for n > 0.
%e A350512 Triangle T(n, k) for 0 <= k <= n starts:
%e A350512 n\k :  0   1    2    3    4    5    6    7   8  9
%e A350512 =================================================
%e A350512   0 :  1
%e A350512   1 :  1   1
%e A350512   2 :  1   4    1
%e A350512   3 :  1   7    7    1
%e A350512   4 :  1  10   18   10    1
%e A350512   5 :  1  13   34   34   13    1
%e A350512   6 :  1  16   55   80   55   16    1
%e A350512   7 :  1  19   81  155  155   81   19    1
%e A350512   8 :  1  22  112  266  350  266  112   22   1
%e A350512   9 :  1  25  148  420  686  686  420  148  25  1
%e A350512   etc.
%t A350512 Flatten[Table[Join[{1},Table[Binomial[n-1,k-1](2*k*(n-k) + n)/k,{k,n}]],{n,0,10}]] (* _Stefano Spezia_, Jan 06 2022 *)
%Y A350512 Row sums are A057711(n+1).
%Y A350512 Cf. A000108, A007318, A086270, A103450.
%K A350512 nonn,easy,tabl
%O A350512 0,5
%A A350512 _Werner Schulte_, Jan 02 2022