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A107105 Triangle, read by rows, where T(n,k) = C(n,k)*(C(n,k) + 1)/2, n>=k>=0.

%I A107105 #20 Jul 20 2025 13:29:14
%S A107105 1,1,1,1,3,1,1,6,6,1,1,10,21,10,1,1,15,55,55,15,1,1,21,120,210,120,21,
%T A107105 1,1,28,231,630,630,231,28,1,1,36,406,1596,2485,1596,406,36,1,1,45,
%U A107105 666,3570,8001,8001,3570,666,45,1,1,55,1035,7260,22155,31878,22155,7260
%N A107105 Triangle, read by rows, where T(n,k) = C(n,k)*(C(n,k) + 1)/2, n>=k>=0.
%C A107105 Replace each number x in Pascal's triangle by x(x+1)/2. - _Charlie Marion_, May 31 2013
%H A107105 Harvey P. Dale, <a href="/A107105/b107105.txt">Table of n, a(n) for n = 0..1000</a>
%F A107105 n-th row sum equals A005317(n) = (C(2n, n) + 2^n)/2.
%e A107105 Triangle begins:
%e A107105 1;
%e A107105 1,1;
%e A107105 1,3,1;
%e A107105 1,6,6,1;
%e A107105 1,10,21,10,1;
%e A107105 1,15,55,55,15,1;
%e A107105 1,21,120,210,120,21,1;
%e A107105 1,28,231,630,630,231,28,1; ...
%t A107105 Table[Binomial[n,k] (Binomial[n,k]+1)/2,{n,0,10},{k,0,n}]//Flatten (* _Harvey P. Dale_, Jul 20 2025 *)
%o A107105 (PARI) T(n,k)=binomial(n,k)*(binomial(n,k)+1)/2
%Y A107105 Cf. A005317 (row sums), A107597 (antidiagonal sums).
%K A107105 nonn,tabl,easy
%O A107105 0,5
%A A107105 _Paul D. Hanna_, May 21 2005