A110330 Inverse of a number triangle related to the Pell numbers.

1, -2, 1, -2, -4, 1, 0, -6, -6, 1, 0, 0, -12, -8, 1, 0, 0, 0, -20, -10, 1, 0, 0, 0, 0, -30, -12, 1, 0, 0, 0, 0, 0, -42, -14, 1, 0, 0, 0, 0, 0, 0, -56, -16, 1, 0, 0, 0, 0, 0, 0, 0, -72, -18, 1, 0, 0, 0, 0, 0, 0, 0, 0, -90, -20, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -110, -22, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -132, -24, 1

Offset: 0

Author

Paul Barry, Jul 20 2005

Keywords

• easy
• sign
• tabl

Comments

This is the matrix inverse of A110327.

Row sums are A110331. Diagonal sums are A110322. Inverse of A110327. The result can be generalized as follows: The triangle whose columns have e.g.f. (x^k/k!)/(1-a*x-b*x^2) has inverse T(n,k)=if(n=k,1,if(n-k=1,-a*binomial(n,1),if(n-k=2,-2*b*binomial(n,2),0))).

Examples

			Rows begin
1;
-2,1;
-2,-4,1;
0,-6,-6,1;
0,0,-12,-8,1;
0,0,0,-20,-10,1;
0,0,0,0,-30,-12,1;
		

Programs

Mathematica

T[n_, k_] := Which[n == k, 1, n-k == 1, -2*Binomial[n, 1], n-k == 2, -2*Binomial[n, 2], True, 0]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 09 2015 *)

PARI

{(T(n,k) = if(n==k, 1, if(n-k==1, -2*binomial(n, 1), if(n-k==2, -2*binomial(n, 2), 0)))); triangle(nMax) = for (n=0, nMax, for (k=0, n, print1(T(n,k), ", ")); print());} \\ Michel Marcus, Dec 02 2013

PARI

egfxy(n,k) = {x = xx + xx*O(xx^n); y = yy + yy*O(yy^k); n!*polcoeff(polcoeff(exp(x*y)*(1-2*x-x^2), n, xx), k, yy);} \\ Michel Marcus, Dec 02 2013

Formula

T(n,k) = if(n=k, 1, if(n-k=1, -2*binomial(n, 1), if(n-k=2, -2*binomial(n, 2), 0))).

E.g.f.: exp(x*y)(1-2x-x^2). This implies that the row polynomials form an Appell sequence. - Tom Copeland, Dec 02 2013