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A205457 Symmetric matrix, by antidiagonals: C(max(2i,2j),min(2i,2j)), i>=0, j>=0.

%I A205457 #11 Nov 15 2020 12:59:26
%S A205457 1,1,1,1,1,1,1,6,6,1,1,15,1,15,1,1,28,15,15,28,1,1,45,70,1,70,45,1,1,
%T A205457 66,210,28,28,210,66,1,1,91,495,210,1,210,495,91,1,1,120,1001,924,45,
%U A205457 45,924,1001,120,1,1,153,1820,3003,495,1,495,3003,1820,153,1,1
%N A205457 Symmetric matrix, by antidiagonals: C(max(2i,2j),min(2i,2j)), i>=0, j>=0.
%F A205457 G.f.: (S(x,y)+S(y,x))/(x*y)-x*y/(1-x*y)+1/(1-x)+1/(1-y)-1, where S(x,y)=((x^3-3*x^2)*y^3-x^2*y^2)/((x^2-2*x+1)*y^3+(-x^2-3)*y^2+(2*x+3)*y-1). - _Vladimir Kruchinin_, Oct 29 2020
%e A205457 Northwest corner:
%e A205457 1....6....15...28...45
%e A205457 6....1....15...70...210
%e A205457 15...15...1....28...210
%e A205457 28...70...28...1....45
%e A205457 45...210..210..45...1
%t A205457 f[i_, j_] := Binomial[Max[2 i - 2, 2 j - 2], Min[2 i - 2, 2 j - 2]]
%t A205457 TableForm[Table[f[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A205457 Flatten[Table[f[i, n + 1 - i], {n, 1, 14}, {i, 1, n}]]
%o A205457 (Maxima)
%o A205457 S(x,y):=((x^3-3*x^2)*y^3-x^2*y^2)/((x^2-2*x+1)*y^3+(-x^2-3)*y^2+(2*x+3)*y-1);
%o A205457 taylor((S(x,y)+S(y,x))/(x*y)-x*y/(1-x*y)+1/(1-x)+1/(1-y)-1,x,0,7,y,0,7); /* _Vladimir Kruchinin_, Oct 29 2020 */
%Y A205457 Cf. A182878, A205456.
%K A205457 nonn,tabl
%O A205457 1,8
%A A205457 _Clark Kimberling_, Jan 28 2012