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Matrix product of Binomial triangle A007318 and triangle with (1, 1, 2, 3, 4, 5, ...) in the main diagonal and the rest zeros.
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%I #16 May 03 2024 08:27:40

%S 1,1,1,1,2,2,1,3,6,3,1,4,12,12,4,1,5,20,30,20,5,1,6,30,60,60,30,6,1,7,

%T 42,105,140,105,42,7,1,8,56,168,280,280,168,56,8,1,9,72,252,504,630,

%U 504,252,72,9

%N Matrix product of Binomial triangle A007318 and triangle with (1, 1, 2, 3, 4, 5, ...) in the main diagonal and the rest zeros.

%C Row sums = A005183: (1, 2, 5, 13, 33, 81, 193, ...).

%F T(n,0) = 1; T(n,k) = k*binomial(n,k) for 1 <= k <= n. [corrected by _Paolo Xausa_, May 03 2024]

%e First few rows of the triangle:

%e 1;

%e 1, 1;

%e 1, 2, 2;

%e 1, 3, 6, 3;

%e 1, 4, 12, 12, 4;

%e 1, 5, 20, 30, 20, 5;

%e 1, 6, 30, 60, 60, 30, 6;

%e 1, 7, 42, 105, 140, 105, 42, 7;

%e ...

%t Table[Max[k, 1]*Binomial[n, k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, May 03 2024 *)

%Y Cf. A005183, A007318.

%K nonn,tabl

%O 0,5

%A _Gary W. Adamson_, Oct 23 2007

%E Corrected and edited by _Olivier GĂ©rard_, Aug 01 2013