A124448 Riordan array (sqrt(1+4x^2)-2x, (1+2x-sqrt(1+4x^2))/2).

1, -2, 1, 2, -3, 1, 0, 4, -4, 1, -2, -1, 7, -5, 1, 0, -4, -4, 11, -6, 1, 4, 2, -6, -10, 16, -7, 1, 0, 8, 8, -6, -20, 22, -8, 1, -10, -5, 11, 19, -1, -35, 29, -9, 1, 0, -20, -20, 7, 34, 13, -56, 37, -10, 1, 28, 14, -26, -46, -12, 49, 41, -84

Offset: 0

Author

Paul Barry, Nov 01 2006

Keywords

• easy
• sign
• tabl

Comments

Inverse of triangle A106195.

Row sums are A105523 (expansion of 1-xc(-x^2) where c(x) is the g.f. of A000108).

Product of A007318 and A124448 is inverse of A053538.

A124448*A007318 = A106180, as infinite lower triangular matrices. - Philippe Deléham, Oct 16 2007

Triangle T(n,k), read by rows, given by (-2,1,-1,1,-1,1,-1,1,-1,...) DELTA (1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 09 2011

Examples

			Triangle begins
   1;
  -2,   1;
   2,  -3,   1;
   0,   4,  -4,   1;
  -2,  -1,   7,  -5,   1;
   0,  -4,  -4,  11,  -6,   1;
   4,   2,  -6, -10,  16,  -7,   1;
   0,   8,   8,  -6, -20,  22,  -8,   1;
		

Crossrefs

Cf. A106195, A000045, A164948, A164942.

Programs

PARI

N=12;
T(n, k)=sum(i=0, n-k, binomial(k, i)*binomial(n-k, i)*2^(n-k-i));
M=matrix(N, N);
for(n=1, N, for(k=1, n, M[n, k]=T(n-1, k-1))); /* A106195 */
A=M^-1;  /* A124448 */
/* for (n=1, N, for(k=1, n, print1(M[n, k], ", "))); */ /* A106195 */
for (n=1, N, for(k=1, n, print1(A[n, k], ", "))); /* A124448 */
/* Joerg Arndt, May 14 2011 */

Extensions

Edited by N. J. A. Sloane, Dec 29 2011