editing

approved

(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)))}

approved

editing

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 (pauldhanna(AT)juno.com), _, Jun 21 2006

(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 (pauldhanna(AT)juno.com), _, Jun 21 2006

_Paul D. Hanna (pauldhanna(AT)juno.com), _, Nov 15 2005

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 (pauldhanna(AT)juno.com), Jun 21 2006

(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 (pauldhanna(AT)juno.com), Jun 21 2006

nonn,tabl,new

Paul D . Hanna (pauldhanna(AT)juno.com), Nov 15 2005

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 (pauldhanna(AT)juno.com), Jun 21 2006

Where 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 + 166*x^2 + 1*x^3)/(1-x)^4/(1-x^2)^3 = 1 + 10*x + 43*x^2 + 31135*x^3 + 51345*x^4 + 76771*x^5 +...

column 34: (1 + 610*x + 618*x^2 + 110*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 +...

= 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 +...

(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 (pauldhanna(AT)juno.com), Jun 21 2006

Cf. A114173 A114177 (row sums), A114174 (central terms), A114175 (row sums-square).

nonn,tabl,new

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

0,5

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 + 16*x^2 + 31*x^3 + 51*x^4 + 76*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 +...

(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)))}

Cf. A114173 (row sums), A114174 (central terms), A114175 (row sums-square).

nonn,tabl

Paul D Hanna (pauldhanna(AT)juno.com), Nov 15 2005

approved