A102916 - OEIS

%I #4 Mar 30 2012 18:36:44

%S 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 125,36,7,1036,6200,9720,3904,1375,216,49,8,5711,61120,69687,47040,

%U 11375,2808,343,64,9,56355,442552,857466,416704,163500,27432,5145,512

%N 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 Column 0 is A102917, the interleaving of A082162 with A102921. Under matrix cube, triangle A102098 shifts each column up 1 row.

%F 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 Rows begin:

%e [1],

%e [1,2],

%e [1,4,3],

%e [3,8,9,4],

%e [7,40,27,16,5],

%e [36,152,189,64,25,6],

%e [139,1128,999,576,125,36,7],

%e [1036,6200,9720,3904,1375,216,49,8],

%e [5711,61120,69687,47040,11375,2808,343,64,9],...

%e The antidiagonals are formed by interleaving the

%e rows of triangle A102098:

%e [1],

%e [1,2],

%e [7,8,3],

%e [139,152,27,4],...

%e with the rows of the matrix square of A102098,

%e which is triangle A102920:

%e [1],

%e [3,4],

%e [36,40,9],

%e [1036,1128,189,16],...

%e G.f. for Column 0 (A102917): 1 = 1*(1-x) + 1*x*(1-x)

%e + 1*x^2*(1-x)(1-2x) + 3*x^3*(1-x)(1-2x)

%e + 7*x^4*(1-x)(1-2x)(1-3x) + 36*x^5*(1-x)(1-2x)(1-3x) +...

%e + A082162(n)*x^(2n)*(1-x)(1-2x)*..*(1-(n+1)x)

%e + A102921(n)*x^(2n+1)*(1-x)(1-2x)*..*(1-(n+1)x) + ...

%e G.f. for Column 1 (A102918): 2 = 2*(1-2x) + 4*x*(1-2x)

%e + 8*x^2*(1-2x)(1-3x) + 40*x^3*(1-2x)(1-3x)

%e + 152*x^4*(1-2x)(1-3x)(1-4x) + 1128*x^5*(1-2x)(1-3x)(1-4x) +...

%e + T(2n+1,1)*x^(2n)*(1-2x)(1-3x)*..*(1-(n+2)x)

%e + T(2n+2,1)*x^(2n+1)*(1-2x)(1-3x)*..*(1-(n+2)x) + ...

%o (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 Cf. A102086, A102098, A102920, A102917, A102918.

%K nonn,tabl

%O 0,3

%A _Paul D. Hanna_, Jan 21 2005