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 A129779 a(1) = 1, a(2) = -1, a(3) = 2; for n > 3, a(n) = -(2*n-5)*a(n-1). 3
 1, -1, 2, -6, 30, -210, 1890, -20790, 270270, -4054050, 68918850, -1309458150, 27498621150, -632468286450, 15811707161250, -426916093353750, 12380566707258750, -383797567925021250, 12665319741525701250 (list; graph; refs; listen; history; text; internal format)
 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: A211211 A127482 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

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Last modified January 22 08:48 EST 2022. Contains 350481 sequences. (Running on oeis4.)