1,3

a(n) = (-1)^(n-1)*A097801(n-2) = (-1)^(n-1)*(2*(n-2))!/((n-2)!*2^(n-3)) for n > 2.

Sequence is also the first column of the inverse of the infinite lower triangular matrix M, where M(j,k) = 1+2*(k-1)*(j-k) for k < j, M(j,k) = 1 for k = j, M(j,k) = 0 for k > j.

Upper left 6x6 submatrix of M is

[1 0 0 0 0 0]

[1 1 0 0 0 0]

[1 3 1 0 0 0]

[1 5 5 1 0 0]

[1 7 9 7 1 0]

[1 9 13 13 9 1],

and upper left 6x6 submatrix of M^-1 is

[ 1 0 0 0 0 0]

[ -1 1 0 0 0 0]

[ 2 -3 1 0 0 0]

[ -6 10 -5 1 0 0]

[ 30 -50 26 -7 1 0]

[ -210 350 -182 50 -9 1].

Row sums of M are 1, 2, 5, 12, 25, 46, ... (see A116731); diagonal sums of M are 1, 1, 2, 4, 7, 13, 20, 32, 45, 65, 86, 116, 147, 189, ... with first differences 0, 1, 2, 3, 6, 7, 12, 13, 20, 21, 30, 31, 42, ... and second differences 1, 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 11, ... (see A093178).

a[n_] := -(2 n - 5) a[n - 1]; a[1] = 1; a[2] = -1; a[3] = 2; Array[a, 20] (* Robert G. Wilson v *)

(PARI) {m=19; print1(1, ", ", -1, ", "); print1(a=2, ", "); for(n=4, m, k=-(2*n-5)*a; print1(k, ", "); a=k)} /* Klaus Brockhaus, May 21 2007 */

(PARI) {print1(1, ", ", -1, ", "); for(n=3, 19, print1((-1)^(n-1)*(2*(n-2))!/((n-2)!*2^(n-3)), ", "))} /* Klaus Brockhaus, May 21 2007 */

(PARI) {m=19; M=matrix(m, m, j, k, if(k>j, 0, if(k==j, 1, 1+2*(k-1)*(j-k)))); print((M^-1)[, 1]~)} /* Klaus Brockhaus, May 21 2007 */

(MAGMA) m:=19; M:=Matrix(IntegerRing(), m, m, [< j, k, Maximum(0, 1+2*(k-1)*(j-k)) > : j, k in [1..m] ] ); Transpose(ColumnSubmatrix(M^-1, 1, 1)); /* Klaus Brockhaus, May 21 2007 */

Cf. A097801, A116731, A093178.

Sequence in context: A104561 A127482 A118747 this_sequence A068215 A096775 A002110

Adjacent sequences: A129776 A129777 A129778 this_sequence A129780 A129781 A129782

sign

Paul Curtz (bpcrtz(AT)free.fr), May 17 2007

Edited and extended by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de) and Robert G. Wilson v (rgwv(AT)rgwv.com), May 21 2007