A121436 Matrix inverse of triangle A122176, where A122176(n,k) = C( k*(k+1)/2 + n-k + 1, n-k) for n>=k>=0.

1, -2, 1, 3, -3, 1, -7, 9, -5, 1, 26, -37, 25, -8, 1, -141, 210, -155, 60, -12, 1, 1034, -1575, 1215, -516, 126, -17, 1, -9693, 14943, -11806, 5270, -1426, 238, -23, 1, 111522, -173109, 138660, -63696, 18267, -3417, 414, -30, 1, -1528112, 2381814, -1923765, 899226, -267084, 53431, -7337, 675, -38, 1

Offset

0,2

Links

Paul D. Hanna, Rows n=0..45, as a table of n, a(n) for n=0..1080.

Formula

(1) T(n,k) = A121435(n-1,k) - A121435(n-1,k+1).

(2) T(n,k) = (-1)^(n-k)*[A107876^(k*(k+1)/2 + 2)](n,k);

i.e., column k equals signed column k of A107876^(k*(k+1)/2 + 2).

G.f.s for column k:

(3) 1 = Sum_{j>=0} T(j+k,k)*x^j/(1-x)^( j*(j+1)/2) + j*k + k*(k+1)/2 + 2);

(4) 1 = Sum_{j>=0} T(j+k,k)*x^j*(1+x)^( j*(j-1)/2) + j*k + k*(k+1)/2 + 2).

Example

Triangle begins:
1;
-2, 1;
3, -3, 1;
-7, 9, -5, 1;
26, -37, 25, -8, 1;
-141, 210, -155, 60, -12, 1;
1034, -1575, 1215, -516, 126, -17, 1;
-9693, 14943, -11806, 5270, -1426, 238, -23, 1;
111522, -173109, 138660, -63696, 18267, -3417, 414, -30, 1;
-1528112, 2381814, -1923765, 899226, -267084, 53431, -7337, 675, -38, 1; ...

Prog

(PARI) /* Matrix Inverse of A122176 */
{T(n, k)=local(M=matrix(n+1, n+1, r, c, if(r>=c, binomial((c-1)*(c-2)/2+r, r-c)))); return((M^-1)[n+1, k+1])}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
(PARI) /* Obtain by G.F. */
{T(n, k)=polcoeff(1-sum(j=0, n-k-1, T(j+k, k)*x^j/(1-x+x*O(x^n))^(j*(j+1)/2+j*k+k*(k+1)/2+2)), n-k)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A098568, A107876; unsigned columns: A107881, A107886.

Sequence in context: A353585 A103525 A294432 * A330694 A088074 A071463

Adjacent sequences: A121433 A121434 A121435 * A121437 A121438 A121439

Keyword

sign,tabl

Author

Paul D. Hanna, Aug 27 2006

Status

approved