A107889 Triangular matrix T, read by rows, that satisfies: [T^-k](n,k) = -T(n,k-1) for n>=k>0, or, equivalently, (column k of T^-k) = -SHIFT_LEFT(column k-1 of T) when zeros above the diagonal are ignored. Also, matrix inverse of triangle A107876.

1, -1, 1, 0, -1, 1, 0, -1, -1, 1, 0, -3, -2, -1, 1, 0, -15, -9, -3, -1, 1, 0, -106, -61, -18, -4, -1, 1, 0, -975, -550, -154, -30, -5, -1, 1, 0, -11100, -6195, -1689, -310, -45, -6, -1, 1, 0, -151148, -83837, -22518, -4005, -545, -63, -7, -1, 1, 0, -2401365, -1326923, -353211, -61686, -8105, -875, -84, -8, -1, 1

Offset

0,12

Comments

SHIFT_LEFT(column 1) = -A107878. SHIFT_LEFT(column 2) = -A107883. SHIFT_LEFT(column 2) = -A107888.

Links

Table of n, a(n) for n=0..65.

Formula

G.f. for column k: 1 = Sum_{j>=0} T(k+j, k)*x^j*(1-x)^(-1 + (k+j)*(k+j-1)/2 - k*(k-1)/2).

Example

G.f. for column 1:
1 = T(1,1)*(1-x)^-1 + T(2,1)*x*(1-x)^0 + T(3,1)*x^2*(1-x)^2 +
T(4,1)*x^3*(1-x)^5 + T(5,1)*x^4*(1-x)^9 + T(6,1)*x^5*(1-x)^14 +...
= 1*(1-x)^-1 - 1*x*(1-x)^0 - 1*x^2*(1-x)^2 - 3*x^3*(1-x)^5 -
15*x^4*(1-x)^9 - 106*x^5*(1-x)^14 - 975*x^6*(1-x)^20 +...
G.f. for column 2:
1 = T(2,2)*(1-x)^-1 + T(3,2)*x*(1-x)^1 + T(4,2)*x^2*(1-x)^4 +
T(5,2)*x^3*(1-x)^8 + T(6,2)*x^4*(1-x)^13 + T(7,2)*x^5*(1-x)^19 +...
= 1*(1-x)^-1 - 1*x*(1-x)^1 - 2*x^2*(1-x)^4 - 9*x^3*(1-x)^8 -
61*x^4*(1-x)^13 - 550*x^5*(1-x)^19 - 6195*x^6*(1-x)^26 +...
Triangle begins:
1;
-1,1;
0,-1,1;
0,-1,-1,1;
0,-3,-2,-1,1;
0,-15,-9,-3,-1,1;
0,-106,-61,-18,-4,-1,1;
0,-975,-550,-154,-30,-5,-1,1;
0,-11100,-6195,-1689,-310,-45,-6,-1,1; ...

Prog

(PARI) {T(n, k)=polcoeff(1-sum(j=0, n-k-1, T(j+k, k)*x^j*(1-x+x*O(x^n))^(-1+(k+j)*(k+j-1)/2-k*(k-1)/2)), n-k)}

Crossrefs

Cf. A107876, A107880, A107884.

Sequence in context: A035103 A155033 A283417 * A138384 A129172 A355693

Adjacent sequences: A107886 A107887 A107888 * A107890 A107891 A107892

Keyword

sign,tabl

Author

Paul D. Hanna, Jun 05 2005

Status

approved