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A102916 Triangle, read by rows, where the antidiagonals are formed by interleaving the rows of triangle A102098 with the rows of its matrix square (A102920).

%I A102916 #4 Mar 30 2012 18:36:44
%S A102916 1,1,2,1,4,3,3,8,9,4,7,40,27,16,5,36,152,189,64,25,6,139,1128,999,576,
%T A102916 125,36,7,1036,6200,9720,3904,1375,216,49,8,5711,61120,69687,47040,
%U A102916 11375,2808,343,64,9,56355,442552,857466,416704,163500,27432,5145,512
%N A102916 Triangle, read by rows, where the antidiagonals are formed by interleaving the rows of triangle A102098 with the rows of its matrix square (A102920).
%C A102916 Column 0 is A102917, the interleaving of A082162 with A102921. Under matrix cube, triangle A102098 shifts each column up 1 row.
%F A102916 G.f. for column k: T(k, k) = k+1 = Sum_{n>=0} T(n+k, k)*x^n*Product_{j=k..[n/2+k]} (1-(j+1)*x).
%e A102916 Rows begin:
%e A102916 [1],
%e A102916 [1,2],
%e A102916 [1,4,3],
%e A102916 [3,8,9,4],
%e A102916 [7,40,27,16,5],
%e A102916 [36,152,189,64,25,6],
%e A102916 [139,1128,999,576,125,36,7],
%e A102916 [1036,6200,9720,3904,1375,216,49,8],
%e A102916 [5711,61120,69687,47040,11375,2808,343,64,9],...
%e A102916 The antidiagonals are formed by interleaving the
%e A102916 rows of triangle A102098:
%e A102916 [1],
%e A102916 [1,2],
%e A102916 [7,8,3],
%e A102916 [139,152,27,4],...
%e A102916 with the rows of the matrix square of A102098,
%e A102916 which is triangle A102920:
%e A102916 [1],
%e A102916 [3,4],
%e A102916 [36,40,9],
%e A102916 [1036,1128,189,16],...
%e A102916 G.f. for Column 0 (A102917): 1 = 1*(1-x) + 1*x*(1-x)
%e A102916 + 1*x^2*(1-x)(1-2x) + 3*x^3*(1-x)(1-2x)
%e A102916 + 7*x^4*(1-x)(1-2x)(1-3x) + 36*x^5*(1-x)(1-2x)(1-3x) +...
%e A102916 + A082162(n)*x^(2n)*(1-x)(1-2x)*..*(1-(n+1)x)
%e A102916 + A102921(n)*x^(2n+1)*(1-x)(1-2x)*..*(1-(n+1)x) + ...
%e A102916 G.f. for Column 1 (A102918): 2 = 2*(1-2x) + 4*x*(1-2x)
%e A102916 + 8*x^2*(1-2x)(1-3x) + 40*x^3*(1-2x)(1-3x)
%e A102916 + 152*x^4*(1-2x)(1-3x)(1-4x) + 1128*x^5*(1-2x)(1-3x)(1-4x) +...
%e A102916 + T(2n+1,1)*x^(2n)*(1-2x)(1-3x)*..*(1-(n+2)x)
%e A102916 + T(2n+2,1)*x^(2n+1)*(1-2x)(1-3x)*..*(1-(n+2)x) + ...
%o A102916 (PARI) {T(n,k)=if(n<k,0,if(n==k,k+1, polcoeff(k+1-sum(i=k,n-1,T(i,k)*x^i*prod(j=1,(i-k)\2+1,1-(j+k)*x+x*O(x^n))),n)))}
%Y A102916 Cf. A102086, A102098, A102920, A102917, A102918.
%K A102916 nonn,tabl
%O A102916 0,3
%A A102916 _Paul D. Hanna_, Jan 21 2005