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

Author

Paul Curtz, May 17 2007

Keywords

• sign

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

Crossrefs

Cf. A093178, A097801, A116731.

Programs

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

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

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 *)

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

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

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

Extensions

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