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Triangle T(n, k) = n*C(n,k) - C(n-1,k-1), 1 <= k <= n, read by rows.
2

%I #10 Feb 01 2022 15:26:15

%S 0,3,1,8,7,2,15,21,13,3,24,46,44,21,4,35,85,110,80,31,5,48,141,230,

%T 225,132,43,6,63,217,427,525,413,203,57,7,80,316,728,1078,1064,700,

%U 296,73,8,99,441,1164,2016,2394,1974,1116,414,91,9,120,595,1770

%N Triangle T(n, k) = n*C(n,k) - C(n-1,k-1), 1 <= k <= n, read by rows.

%C Mirror of A208657.

%C col 1: A005563

%C col 2: A127736

%C top edge: A000027

%e First five rows:

%e 0

%e 3....1

%e 8....7....2

%e 15...21...13...3

%e 24...46...44...21...4

%t z = 12;

%t f[n_, k_] := n*Binomial[n, k] - Binomial[n - 1, k - 1]

%t t = Table[f[n, k], {n, 1, z}, {k, 1, n}];

%t TableForm[t] (* A208656 as a triangle *)

%t Flatten[t] (* A208656 as a sequence *)

%t r = Table[f[n, k], {n, 1, z}, {k, n, 1, -1}];

%t TableForm[r] (* A208657 as a triangle *)

%t Flatten[r] (* A208657 as a sequence *)

%t Table[Sum[f[n, k], {k, 1, n}], {n, 1, 3 z}](* A208658 *)

%Y Cf. A208657, A208658.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 01 2012

%E Definition amended by _Georg Fischer_, Feb 01 2022