A200171 Triangle, read by rows, such that row n equals the coefficients of x^(n^2+n-1+k) in F(x,n) for k = 1..n, where F(x,n) = (1 + x*F(x,n))*(1 + x^n/F(x,n)), for n>=1.

1, 4, 1, -3, 7, 1, 16, -19, 11, 1, -40, 86, -54, 16, 1, 134, -328, 302, -118, 22, 1, -427, 1289, -1483, 827, -223, 29, 1, 1432, -5003, 7009, -5003, 1927, -383, 37, 1, -4860, 19450, -32030, 28030, -14012, 4006, -614, 46, 1, 16798, -75580, 143210, -148510, 91730, -34396, 7646, -934, 56, 1

Offset

1,2

Comments

The initial n rows of this triangle are obtained from the initial (n+1)^2 - 1 coefficients of the function F(x,n) = (1 + x*F(x,n))*(1 + x^n/F(x,n)) upon removing the n leading 1's and thereafter removing 2's; see the example section for illustrations of this pattern.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..465

Formula

T(n,1) = (-1)^n*A000108(n) + 2 = (-1)^n*binomial(2*n,n)/(n+1) + 2, for n>=1.

T(n,2) = (-1)^(n-1)*binomial(2*n-1,n-2) + 2, for n>=2.

Example

Triangle begins:
1;
4, 1;
-3, 7, 1;
16, -19, 11, 1;
-40, 86, -54, 16, 1;
134, -328, 302, -118, 22, 1;
-427, 1289, -1483, 827, -223, 29, 1;
1432, -5003, 7009, -5003, 1927, -383, 37, 1;
-4860, 19450, -32030, 28030, -14012, 4006, -614, 46, 1;
16798, -75580, 143210, -148510, 91730, -34396, 7646, -934, 56, 1;
-58784, 293932, -629848, 755822, -556918, 259898, -76438, 13652, -1363, 67, 1;
208014, -1144064, 2735812, -3730648, 3197702, -1790710, 659738, -157078, 23102, -1923, 79, 1; ...
Row sums begin: [1,5,5,9,9,13,13,17,17,21,21,25,25,29,29,...].
ILLUSTRATION OF INITIAL ROWS.
The rows of this triangle can be generated in the following manner.
For row 7, the coefficients in F(x,7) = (1 + x*F(x,7))*(1 + x^7/F(x,7)) begin:
[1,1,1,1,1,1,1, 2,2,2,2,2,2,2, 1, 2,2,2,2,2,2, 4,1, 2,2,2,2,2, -3,7,1, 2,2,2,2, 16,-19,11,1, 2,2,2, -40,86,-54,16,1, 2,2, 134,-328,302,-118,22,1, 2, -427,1289,-1483,827,-223,29,1, ...],
which can be arranged like so:
1,1,1,1,1,1,1,
2,2,2,2,2,2,2,
1,
2,2,2,2,2,2,
4,1,
2,2,2,2,2,
-3,7,1,
2,2,2,2,
16,-19,11,1,
2,2,2,
-40,86,-54,16,1,
2,2,
134,-328,302,-118,22,1,
2,
-427,1289,-1483,827,-223,29,1, ...;
then, if we remove all 2's and the first row of 1's, we obtain the initial 7 rows of this triangle.
This triangle is the limit of the above process.

Prog

(PARI) {T(n, k)=local(A=1+x); for(i=1, n^2+n+k, A=(1+x*A)*(1+x^n/(A+x*O(x^(n^2+k))))); polcoeff(A, n^2+n-1+k)}
{for(n=1, 15, for(k=1, n, print1(T(n, k), ", ")); print(""))}

Crossrefs

Cf. A200172 (column 3), A200173 (column 4), A200140 (central terms).

Sequence in context: A074813 A151861 A210583 * A109531 A200132 A073817

Adjacent sequences: A200168 A200169 A200170 * A200172 A200173 A200174

Keyword

sign,tabl

Author

Paul D. Hanna, Nov 13 2011

Status

approved