A129779 a(1) = 1, a(2) = -1, a(3) = 2; for n > 3, a(n) = -(2*n-5)*a(n-1).

1, -1, 2, -6, 30, -210, 1890, -20790, 270270, -4054050, 68918850, -1309458150, 27498621150, -632468286450, 15811707161250, -426916093353750, 12380566707258750, -383797567925021250, 12665319741525701250

Offset

1,3

Comments

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 6 X 6 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 6 X 6 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).

Links

G. C. Greubel, Table of n, a(n) for n = 1..400

Formula

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

G.f.: 1 + x - x*W(0) , where W(k) = 1 + 1/( 1 - x*(2*k+1)/( x*(2*k+1) - 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 22 2013

Maple

seq(`if`(n<3, (-1)^(n-1), (-1)^(n-1)*(2*n-5)!/(2^(n-4)*(n-3)!)), n=1..25); # G. C. Greubel, Nov 25 2019

Mathematica

a[n_]:= -(2*n-5)*a[n-1]; a[1]=1; a[2]=-1; a[3]=2; Array[a, 20] (* Robert G. Wilson v *)
Table[If[n<3, (-1)^(n-1), (-1)^(n+1)*(2*n-5)!/(2^(n-4)*(n-3)!)], {n, 25}] (* G. C. Greubel, Nov 25 2019 *)

Prog

(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
(Magma) F:=Factorial; [1, -1] cat [(-1)^(n+1)*F(2*n-5)/(2^(n-4)*F(n-3)): n in [3..25]]; // G. C. Greubel, Nov 25 2019
(Sage) f=factorial; [1, -1]+[(-1)^(n+1)*f(2*n-5)/(2^(n-4)*f(n-3)) for n in (3..25)] # G. C. Greubel, Nov 25 2019
(GAP) F:=Factorial;; Concatenation([1, -1], List([3..25], n-> (-1)^(n+1)*F(2*n-5)/(2^(n-4)*F(n-3)) )); # G. C. Greubel, Nov 25 2019

Crossrefs

Cf. A093178, A097801, A116731.

Sequence in context: A127482 A372741 A118747 * A068215 A305400 A096775

Adjacent sequences: A129776 A129777 A129778 * A129780 A129781 A129782

Keyword

sign

Author

Paul Curtz, May 17 2007

Extensions

Edited and extended by Klaus Brockhaus and Robert G. Wilson v, May 21 2007

Status

approved