A114172 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), for n>=0; e.g., C_n(x) = R_n(x)/(1-x)^(n+1).

1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 7, 16, 9, 1, 1, 9, 31, 36, 12, 1, 1, 11, 51, 95, 66, 15, 1, 1, 13, 76, 199, 229, 106, 18, 1, 1, 15, 106, 361, 601, 467, 156, 21, 1, 1, 17, 141, 594, 1316, 1509, 844, 216, 24, 1, 1, 19, 181, 911, 2542, 3951, 3293, 1395, 286, 27, 1, 1, 21, 226

Offset

0,5

Links

Table of n, a(n) for n=0..68.

Example

Triangle begins:
1;
1,1;
1,3,1;
1,5,6,1;
1,7,16,9,1;
1,9,31,36,12,1;
1,11,51,95,66,15,1;
1,13,76,199,229,106,18,1;
1,15,106,361,601,467,156,21,1;
1,17,141,594,1316,1509,844,216,24,1;
1,19,181,911,2542,3951,3293,1395,286,27,1;
1,21,226,1325,4481,8910,10193,6447,2155,366,30,1; ...
Where g.f. for columns is formed from g.f. of rows:
GF(column 2) = (1 + 3*x + 1*x^2)/(1-x)^3
= 1 + 6*x + 16*x^2 + 31*x^3 + 51*x^4 + 76*x^5 +...
GF(column 3) = (1 + 5*x + 6*x^2 + 1*x^3)/(1-x)^4
= 1 + 9*x + 36*x^2 + 95*x^3 + 199*x^4 + 361*x^5 +...
GF(column 4) = (1 + 7*x + 16*x^2 + 9*x^3 + 1*x^4)/(1-x)^5
= 1 + 12*x + 66*x^2 + 229*x^3 + 601*x^4 + 1316*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), n-k)))}

Crossrefs

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

Sequence in context: A202672 A054142 A076756 * A271942 A121522 A294582

Adjacent sequences: A114169 A114170 A114171 * A114173 A114174 A114175

Keyword

nonn,tabl

Author

Paul D. Hanna, Nov 15 2005

Status

approved