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A144480 T(n,k) = binomial(n, k)*min(k + 1, n - k + 1), triangle read by rows (n >= 0, 0 <= k <= n).

%I A144480 #15 Jan 06 2019 15:51:45
%S A144480 1,1,1,1,4,1,1,6,6,1,1,8,18,8,1,1,10,30,30,10,1,1,12,45,80,45,12,1,1,
%T A144480 14,63,140,140,63,14,1,1,16,84,224,350,224,84,16,1,1,18,108,336,630,
%U A144480 630,336,108,18,1,1,20,135,480,1050,1512,1050,480,135,20,1
%N A144480 T(n,k) = binomial(n, k)*min(k + 1, n - k + 1), triangle read by rows (n >= 0, 0 <= k <= n).
%F A144480 If k <= floor(n/2), then T(n,k) = binomial(n, k)*(k + 1), otherwise T(n,k) = binomial(n, k)*(n - k - 1).
%F A144480 T(n,k) = A007318(n,k)*A003983(k+1,n-k+1), i.e., term-by term product of Pascal's triangle A007318 and A003983 as a triangle.
%e A144480 Triangle begins:
%e A144480   1;
%e A144480   1,  1;
%e A144480   1,  4,   1;
%e A144480   1,  6,   6,   1;
%e A144480   1,  8,  18,   8,    1;
%e A144480   1, 10,  30,  30,   10,    1;
%e A144480   1, 12,  45,  80,   45,   12,    1;
%e A144480   1, 14,  63, 140,  140,   63,   14,   1;
%e A144480   1, 16,  84, 224,  350,  224,   84,  16,   1;
%e A144480   1, 18, 108, 336,  630,  630,  336, 108,  18,  1;
%e A144480   1, 20, 135, 480, 1050, 1512, 1050, 480, 135, 20, 1;
%e A144480   ...
%t A144480 Table[Table[Binomial[n, m]*If[m <= Floor[n/2], 1 + m, 1 + n - m], {m, 0, n}], {n, 0, 10}] // Flatten
%o A144480 (Maxima) create_list(binomial(n, k)*min(k + 1, n - k + 1), n, 0, 10, k, 0, n); /* _Franck Maminirina Ramaharo_, Dec 10 2018 */
%Y A144480 Row sums are in A245560.
%Y A144480 Cf. A003983, A007318, A168643.
%K A144480 nonn,easy,tabl
%O A144480 0,5
%A A144480 _Roger L. Bagula_, Oct 11 2008
%E A144480 Entry revised by _N. J. A. Sloane_, Aug 07 2014
%E A144480 Edited by _Franck Maminirina Ramaharo_, Dec 10 2018