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A104029 Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).

%I A104029 #10 Feb 06 2013 12:25:52
%S A104029 1,2,1,3,5,1,4,13,9,1,5,26,35,14,1,6,45,96,75,20,1,7,71,216,267,140,
%T A104029 27,1,8,105,427,750,623,238,35,1,9,148,770,1800,2123,1288,378,44,1,10,
%U A104029 201,1296,3858,6046,5211,2436,570,54,1,11,265,2067,7590,15115,17303,11505
%N A104029 Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).
%C A104029 Matrix inverse is A104030. Antidiagonal sums form unsigned A078039.
%F A104029 G.f.: A(x, y) = (1-x*y)/(1 - 2*x*(1+y) + x^2*(1+y+y^2) ).
%F A104029 T(n, k) = [x^(2k)](1+x+x^2)^n + [x^(2k+1)](1+x+x^2)^n.
%e A104029 Row 3: {4,13,9,1} is formed from the pairwise sums
%e A104029 of row 3 of A027907: {1,3, 6,7, 6,3, 1}.
%e A104029 Rows begin:
%e A104029 1;
%e A104029 2, 1;
%e A104029 3, 5, 1;
%e A104029 4, 13, 9, 1;
%e A104029 5, 26, 35, 14, 1;
%e A104029 6, 45, 96, 75, 20, 1;
%e A104029 7, 71, 216, 267, 140, 27, 1;
%e A104029 8, 105, 427, 750, 623, 238, 35, 1;
%e A104029 9, 148, 770, 1800, 2123, 1288, 378, 44, 1;
%e A104029 10, 201, 1296, 3858, 6046, 5211, 2436, 570, 54, 1;
%e A104029 11, 265, 2067, 7590, 15115, 17303, 11505, 4302, 825, 65, 1;
%e A104029 12, 341, 3157, 13959, 34210, 49721, 43923, 23397, 7194, 1155, 77, 1; ...
%o A104029 (PARI) {T(n,k)=polcoeff((1+x+x^2)^n+x*O(x^(2*k)),2*k)+ polcoeff((1+x+x^2)^n+x*O(x^(2*k+1)),2*k+1)}
%o A104029 for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%o A104029 (PARI) {T(n,k)=polcoeff(polcoeff((1-x*y)/(1-2*x*(1+y)+x^2*(1+y+y^2)) +x*O(x^n),n,x)+y*O(y^k),k,y)}
%Y A104029 Cf. A104030, A104027, A078039.
%K A104029 nonn,tabl
%O A104029 0,2
%A A104029 _Paul D. Hanna_, Feb 26 2005