A114176 - OEIS

%I #6 Jun 13 2017 23:40:22

%S 1,1,1,1,3,1,1,6,6,1,1,10,18,10,1,1,15,43,43,15,1,1,21,86,135,87,21,1,

%T 1,28,156,345,345,159,28,1,1,36,260,771,1083,777,267,36,1,1,45,410,

%U 1557,2901,2927,1577,423,45,1,1,55,615,2913,6909,9219,7001,2973,637,55,1

%N Triangle, read by rows, where the g.f. of column n, C_n(x), equals the g.f. of row n, R_n(x), divided by (1-x)^(n+1)*(1-x^2)^n, for n>=0; e.g., C_n(x) = R_n(x)/(1-x)^(n+1)/(1-x^2)^n.

%F T(n,k) = Sum_{j=0..k} T(k,j)*Sum_{i=0..n-j-k} (-1)^(n-i-j-k)*C(2k+i,i)*C(n-i-j-1,n-i-j-k) for n>k with T(n,n)=1 for n>=0. - _Paul D. Hanna_, Jun 21 2006

%e Triangle begins:

%e 1;

%e 1,1;

%e 1,3,1;

%e 1,6,6,1;

%e 1,10,18,10,1;

%e 1,15,43,43,15,1;

%e 1,21,86,135,87,21,1;

%e 1,28,156,345,345,159,28,1;

%e 1,36,260,771,1083,777,267,36,1;

%e 1,45,410,1557,2901,2927,1577,423,45,1;

%e 1,55,615,2913,6909,9219,7001,2973,637,55,1; ...

%e where g.f. for columns is formed from g.f. of rows:

%e column 2: (1 + 3*x + 1*x^2)/(1-x)^3/(1-x^2)^2 = 1 + 6*x + 18*x^2 + 43*x^3 + 86*x^4 + 156*x^5 +...

%e column 3: (1 + 6*x + 6*x^2 + 1*x^3)/(1-x)^4/(1-x^2)^3 = 1 + 10*x + 43*x^2 + 135*x^3 + 345*x^4 + 771*x^5 +...

%e column 4: (1 + 10*x + 18*x^2 + 10*x^3 + 1*x^4)/(1-x)^5/(1-x^2)^4 = 1 + 15*x + 87*x^2 + 345*x^3 + 1083*x^4 + 2901*x^5 +...

%o (PARI) T(n,k)=if(n<k || k<0,0,if(n==k || k==0,1, polcoeff(sum(j=0,k,T(k,j)*x^j)/(1-x+x*O(x^(n-k)))^(k+1)/(1-x^2)^k,n-k)))

%o (PARI) {T(n,k)=if(n==k,1,sum(j=0,k,T(k,j)*sum(i=0,n-j-k, (-1)^(n-i-j-k)*binomial(2*k+i,i)*binomial(n-i-j-1,n-i-j-k))))} - _Paul D. Hanna_, Jun 21 2006

%Y Cf. A114177 (row sums), A114174 (central terms), A114175 (row sums-square).

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Nov 15 2005