A121435 Matrix inverse of triangle A122175, where A122175(n,k) = C( k*(k+1)/2 + n-k, n-k) for n>=k>=0.

1, -1, 1, 1, -2, 1, -2, 5, -4, 1, 7, -19, 18, -7, 1, -37, 104, -106, 49, -11, 1, 268, -766, 809, -406, 110, -16, 1, -2496, 7197, -7746, 4060, -1210, 216, -22, 1, 28612, -82910, 90199, -48461, 15235, -3032, 385, -29, 1, -391189, 1136923, -1244891, 678874, -220352, 46732, -6699, 638, -37, 1

Offset

0,5

Links

Table of n, a(n) for n=0..54.

Formula

(1) T(n,k) = A121434(n-1,k) - A121434(n-1,k+1).

(2) T(n,k) = (-1)^(n-k)*[A107876^(k*(k+1)/2 + 1)](n,k); i.e., column k equals signed column k of matrix power A107876^(k*(k+1)/2 + 1).

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 + 1);

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

Example

Triangle begins:
1;
-1, 1;
1, -2, 1;
-2, 5, -4, 1;
7, -19, 18, -7, 1;
-37, 104, -106, 49, -11, 1;
268, -766, 809, -406, 110, -16, 1;
-2496, 7197, -7746, 4060, -1210, 216, -22, 1;
28612, -82910, 90199, -48461, 15235, -3032, 385, -29, 1;
-391189, 1136923, -1244891, 678874, -220352, 46732, -6699, 638, -37, 1; ...

Prog

(PARI) /* Matrix Inverse of A122175 */ T(n, k)=local(M=matrix(n+1, n+1, r, c, if(r>=c, binomial((c-1)*(c-2)/2+r-1, r-c)))); return((M^-1)[n+1, k+1])
(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+1)), n-k)

Crossrefs

Cf. A098568, A107876; unsigned columns: A107877, A107882.

Sequence in context: A201780 A337991 A104560 * A137156 A136457 A209133

Adjacent sequences: A121432 A121433 A121434 * A121436 A121437 A121438

Keyword

sign,tabl

Author

Paul D. Hanna, Aug 27 2006

Status

approved