A114176 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.

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, 1, 28, 156, 345, 345, 159, 28, 1, 1, 36, 260, 771, 1083, 777, 267, 36, 1, 1, 45, 410, 1557, 2901, 2927, 1577, 423, 45, 1, 1, 55, 615, 2913, 6909, 9219, 7001, 2973, 637, 55, 1

Offset

0,5

Links

Table of n, a(n) for n=0..65.

Formula

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

Example

Triangle begins:
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;
1,28,156,345,345,159,28,1;
1,36,260,771,1083,777,267,36,1;
1,45,410,1557,2901,2927,1577,423,45,1;
1,55,615,2913,6909,9219,7001,2973,637,55,1; ...
where g.f. for columns is formed from g.f. of rows:
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 +...
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 +...
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 +...

Prog

(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)))
(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

Crossrefs

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

Sequence in context: A054120 A299146 A362242 * A056241 A162745 A001263

Adjacent sequences: A114173 A114174 A114175 * A114177 A114178 A114179

Keyword

nonn,tabl

Author

Paul D. Hanna, Nov 15 2005

Status

approved