A227964 Triangle where the g.f. of row n equals (1-x-x^2+x^3)^n and terms T(n,k) are read by rows n>=0, k=0..3*n.

1, 1, -1, -1, 1, 1, -2, -1, 4, -1, -2, 1, 1, -3, 0, 8, -6, -6, 8, 0, -3, 1, 1, -4, 2, 12, -17, -8, 28, -8, -17, 12, 2, -4, 1, 1, -5, 5, 15, -35, -1, 65, -45, -45, 65, -1, -35, 15, 5, -5, 1, 1, -6, 9, 16, -60, 24, 116, -144, -66, 220, -66, -144, 116, 24, -60, 16, 9, -6, 1, 1, -7, 14, 14, -91, 77, 168, -344, -14, 546, -364, -364, 546, -14, -344, 168, 77, -91, 14, 14, -7, 1

Offset

0,7

Links

Paul D. Hanna, Table of n, a(n) for n = 0..2500

Formula

Sum_{k=0..3*n} |T(n,k)| = A192205(n).

Sum_{k=0..3*n} T(n,k)^2 = binomial(4*n,n).

Sum_{k=0..3*n} T(n,k) * binomial(3*n,k) = (-1)^n * binomial(4*n,n).

Sum_{k=0..3*n} T(n,k) * binomial(2*n+k,k) = 2^n.

Sum_{k=0..3*n} T(n,k) * binomial(3*n+k,k) = A008288(3*n,n), where A008288 is the Delannoy array (see A026001).

Example

Triangle begins:
1;
1, -1, -1, 1;
1, -2, -1, 4, -1, -2, 1;
1, -3, 0, 8, -6, -6, 8, 0, -3, 1;
1, -4, 2, 12, -17, -8, 28, -8, -17, 12, 2, -4, 1;
1, -5, 5, 15, -35, -1, 65, -45, -45, 65, -1, -35, 15, 5, -5, 1;
1, -6, 9, 16, -60, 24, 116, -144, -66, 220, -66, -144, 116, 24, -60, 16, 9, -6, 1;
1, -7, 14, 14, -91, 77, 168, -344, -14, 546, -364, -364, 546, -14, -344, 168, 77, -91, 14, 14, -7, 1; ...

Prog

(PARI) {T(n, k)=polcoeff((1-x-x^2+x^3 +x*O(x^k))^n, k)}
for(n=0, 10, for(k=0, 3*n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A192205.

Sequence in context: A305333 A127772 A086256 * A057550 A324892 A059150

Adjacent sequences: A227961 A227962 A227963 * A227965 A227966 A227967

Keyword

sign,tabf

Author

Paul D. Hanna, Aug 01 2013

Status

approved