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A172991 Triangle of binomial sums read by rows: T(n,k) = sum(C(2n-2k-i,i) * C(2k-i,i), i=0..min(k,n-k)).

%I A172991 #20 Mar 14 2014 12:39:10
%S A172991 1,1,1,1,2,1,1,4,4,1,1,6,11,6,1,1,8,22,22,8,1,1,10,37,63,37,10,1,1,12,
%T A172991 56,136,136,56,12,1,1,14,79,249,376,249,79,14,1,1,16,106,410,849,849,
%U A172991 410,106,16,1,1,18,137,627,1663,2317,1663,627,137,18,1,1,20,172,908,2942,5371,5371,2942,908,172,20,1,1,22,211,1261,4826,11017,14545,11017,4826,1261,211,22,1
%N A172991 Triangle of binomial sums read by rows: T(n,k) = sum(C(2n-2k-i,i) * C(2k-i,i), i=0..min(k,n-k)).
%C A172991 The matrix inverse starts
%C A172991 1;
%C A172991 -1,1;
%C A172991 1,-2,1;
%C A172991 -1,4,-4,1;
%C A172991 0,-8,13,-6,1;
%C A172991 7,12,-38,26,-8,1;
%C A172991 -35,-12,114,-101,43,-10,1; - _R. J. Mathar_, Mar 22 2013
%F A172991 G.f.: (1 -x -x*y -2*x^2*y +x^3*y +x^3*y^2 +4*x^4*y^2 -x^6*y^3) / (1 -2*x +x^2 -2*x*y+2*x^3*y +x^2*y^2 +2*x^3*y^2 +3*x^4*y^2 -2*x^5*y^2 -2*x^5*y^3 -6*x^6*y^3 +x^8*y^4).
%F A172991 Central coefficients T(2n,n) = A188648.
%e A172991 G.f. =
%e A172991 1 +
%e A172991 (y + 1)*x +
%e A172991 (y^2 + 2*y + 1)*x^2 +
%e A172991 (y^3 + 4*y^2 + 4*y + 1)*x^3 +
%e A172991 (y^4 + 6*y^3 + 11*y^2 + 6*y + 1)*x^4 + ...
%e A172991 Triangle begins:
%e A172991 1,
%e A172991 1,  1,
%e A172991 1,  2,  1,
%e A172991 1,  4,  4,   1,
%e A172991 1,  6, 11,   6,   1,
%e A172991 1,  8, 22,  22,   8,   1,
%e A172991 1, 10, 37,  63,  37,  10,  1,
%e A172991 1, 12, 56, 136, 136,  56, 12,  1,
%e A172991 1, 14, 79, 249, 376, 249, 79, 14,  1
%t A172991 Flatten[Table[Sum[Binomial[2n-2k-i,i]Binomial[2k-i,i],{i,0,Min[k,n-k]}],{n,0,12},{k,0,n}]]
%o A172991 (Maxima) create_list(sum(binomial(2*n-2*k-i,i)*binomial(2*k-i,i),i,0,min(k,n-k)),n,0,10,k,0,n);
%Y A172991 Cf. A188648, A054142, A027989.
%K A172991 nonn,easy,tabl
%O A172991 0,5
%A A172991 _Emanuele Munarini_, Apr 07 2011