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A143418 Triangle read by rows. T(n,k) = binomial(n,k)*(binomial(n,k)-1)/2.

%I A143418 #15 Jul 14 2024 19:43:18
%S A143418 1,3,3,6,15,6,10,45,45,10,15,105,190,105,15,21,210,595,595,210,21,28,
%T A143418 378,1540,2415,1540,378,28,36,630,3486,7875,7875,3486,630,36,45,990,
%U A143418 7140,21945,31626,21945,7140,990,45,55,1485,13530,54285,106491,106491
%N A143418 Triangle read by rows. T(n,k) = binomial(n,k)*(binomial(n,k)-1)/2.
%C A143418 Row sums = A108958: (1, 6, 27, 110, 430, 1652, ...).
%H A143418 Harvey P. Dale, <a href="/A143418/b143418.txt">Table of n, a(n) for n = 1..1000</a>
%F A143418 T(n,k) = A065420(n-1,k-1)/2. - _R. J. Mathar_, Apr 04 2012
%e A143418 Row 4 of Pascal's triangle (1, 4, 6, 4, 1) with each term squared = (1, 16, 36, 16, 1), then subtracting (1, 4, 6, 4, 1) = (0, 12, 30, 12, 0). Dividing by 2 and deleting the zeros, we get row 4 of A143418: (6, 15, 6).
%e A143418 First few rows of the triangle =
%e A143418 1;
%e A143418 3, 3;
%e A143418 6, 15, 6;
%e A143418 10, 45, 45, 10;
%e A143418 15, 105, 190, 105, 15;
%e A143418 21, 210, 595, 595, 210, 21;
%e A143418 28, 378, 1540, 2415, 1540, 378, 28;
%e A143418 ...
%p A143418 A143418 := proc(n,k)
%p A143418         binomial(n,k)*(binomial(n,k)-1)/2 ;
%p A143418 end proc:
%p A143418 seq(seq(A143418(n,k),k=1..n-1),n=1..12) ; # _R. J. Mathar_, Apr 04 2012
%t A143418 Table[Binomial[n,k] (Binomial[n,k]-1)/2,{n,20},{k,n-1}]//Flatten (* _Harvey P. Dale_, Jun 14 2021 *)
%Y A143418 Cf. A007318, A108958.
%K A143418 nonn,easy,tabl
%O A143418 1,2
%A A143418 _Gary W. Adamson_ and _Roger L. Bagula_, Aug 14 2008
%E A143418 Corrected by _Harvey P. Dale_, Jun 14 2021