A107889 - OEIS

%I #16 Jun 11 2024 01:27:25

%S 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,

%T -4,-1,1,0,-975,-550,-154,-30,-5,-1,1,0,-11100,-6195,-1689,-310,-45,

%U -6,-1,1,0,-151148,-83837,-22518,-4005,-545,-63,-7,-1,1,0,-2401365,-1326923,-353211,-61686,-8105,-875,-84,-8,-1,1

%N 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.

%C SHIFT_LEFT(column 1) = -A107878.

%C SHIFT_LEFT(column 2) = -A107883.

%C SHIFT_LEFT(column 3) = -A107888.

%F 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).

%e G.f. for column 1:

%e 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 + ...

%e   = 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 + ...

%e G.f. for column 2:

%e 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 + ...

%e   = 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 + ...

%e Triangle begins:

%e    1;

%e   -1,      1;

%e    0,     -1,     1;

%e    0,     -1,    -1,     1;

%e    0,     -3,    -2,    -1,    1;

%e    0,    -15,    -9,    -3,   -1,   1;

%e    0,   -106,   -61,   -18,   -4,  -1,  1;

%e    0,   -975,  -550,  -154,  -30,  -5, -1,  1;

%e    0, -11100, -6195, -1689, -310, -45, -6, -1, 1;

%e   ...

%t max = 10;

%t A107862 = Table[Binomial[If[n < k, 0, n*(n-1)/2-k*(k-1)/2 + n - k], n - k], {n, 0, max}, {k, 0, max}];

%t A107867 = Table[Binomial[If[n < k, 0, n*(n-1)/2-k*(k-1)/2 + n-k+1], n - k], {n, 0, max}, {k, 0, max}];

%t T = Inverse[Inverse[A107862].A107867];

%t Table[T[[n + 1, k + 1]], {n, 0, max}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 31 2024 *)

%o (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)}

%Y Cf. A107876, A107880, A107884.

%K sign,tabl

%O 0,12

%A _Paul D. Hanna_, Jun 05 2005