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A208657 Triangular array read by rows: n*binomial(n,n-k+1)-binomial(n-1,n-k) with k = 1..n, n >= 1.

%I A208657 #17 Sep 08 2022 08:46:01
%S A208657 0,1,3,2,7,8,3,13,21,15,4,21,44,46,24,5,31,80,110,85,35,6,43,132,225,
%T A208657 230,141,48,7,57,203,413,525,427,217,63,8,73,296,700,1064,1078,728,
%U A208657 316,80,9,91,414,1116,1974,2394,2016,1164,441,99,10,111,560,1695
%N A208657 Triangular array read by rows: n*binomial(n,n-k+1)-binomial(n-1,n-k) with k = 1..n, n >= 1.
%C A208657 Mirror of A208656.
%e A208657 Triangle begins:
%e A208657 0,
%e A208657 1, 3,
%e A208657 2, 7, 8,
%e A208657 3, 13, 21, 15,
%e A208657 4, 21, 44, 46, 24,
%e A208657 5, 31, 80, 110, 85, 35,
%e A208657 6, 43, 132, 225, 230, 141, 48,
%e A208657 7, 57, 203, 413, 525, 427, 217, 63,
%e A208657 8, 73, 296, 700, 1064, 1078, 728, 316, 80,
%e A208657 9, 91, 414, 1116, 1974, 2394, 2016, 1164, 441, 99;
%e A208657 ...
%t A208657 z = 12;
%t A208657 f[n_, k_] := n*Binomial[n, k] - Binomial[n - 1, k - 1]
%t A208657 t = Table[f[n, k], {n, 1, z}, {k, 1, n}];
%t A208657 TableForm[t] (* A208656 as a triangle *)
%t A208657 Flatten[t]   (* A208656 as a sequence *)
%t A208657 r = Table[f[n, k], {n, 1, z}, {k, n, 1, -1}];
%t A208657 TableForm[r] (* A208657 as a triangle *)
%t A208657 Flatten[r]   (* A208657 as a sequence *)
%t A208657 Table[Sum[f[n, k], {k, 1, n}], {n, 1, 3 z}](* A208658 *)
%o A208657 (Magma) [n*Binomial(n,n-k+1)-Binomial(n-1,n-k): k in [1..n], n in [1..11]]; // _Bruno Berselli_, Apr 15 2015
%Y A208657 Cf. A002061 (second column), A208656, A208658 (row sums), A257055.
%K A208657 nonn,tabl,easy
%O A208657 1,3
%A A208657 _Clark Kimberling_, Mar 01 2012