This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A129779 #22 Sep 12 2024 17:49:11
%S A129779 1,-1,2,-6,30,-210,1890,-20790,270270,-4054050,68918850,-1309458150,
%T A129779 27498621150,-632468286450,15811707161250,-426916093353750,
%U A129779 12380566707258750,-383797567925021250,12665319741525701250
%N A129779 a(1) = 1, a(2) = -1, a(3) = 2; for n > 3, a(n) = -(2*n-5)*a(n-1).
%C A129779 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.
%C A129779 Upper left 6 X 6 submatrix of M is
%C A129779 [ 1 0 0 0 0 0 ]
%C A129779 [ 1 1 0 0 0 0 ]
%C A129779 [ 1 3 1 0 0 0 ]
%C A129779 [ 1 5 5 1 0 0 ]
%C A129779 [ 1 7 9 7 1 0 ]
%C A129779 [ 1 9 13 13 9 1 ],
%C A129779 and upper left 6 X 6 submatrix of M^-1 is
%C A129779 [ 1 0 0 0 0 0 ]
%C A129779 [ -1 1 0 0 0 0 ]
%C A129779 [ 2 -3 1 0 0 0 ]
%C A129779 [ -6 10 -5 1 0 0 ]
%C A129779 [ 30 -50 26 -7 1 0 ]
%C A129779 [ -210 350 -182 50 -9 1 ].
%C A129779 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).
%H A129779 G. C. Greubel, <a href="/A129779/b129779.txt">Table of n, a(n) for n = 1..400</a>
%F A129779 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.
%F A129779 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
%p A129779 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
%t A129779 a[n_]:= -(2*n-5)*a[n-1]; a[1]=1; a[2]=-1; a[3]=2; Array[a, 20] (* _Robert G. Wilson v_ *)
%t A129779 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 *)
%o A129779 (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
%o A129779 (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
%o A129779 (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
%o A129779 (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
%o A129779 (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
%o A129779 (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
%o A129779 (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
%Y A129779 Cf. A093178, A097801, A116731.
%K A129779 sign
%O A129779 1,3
%A A129779 _Paul Curtz_, May 17 2007
%E A129779 Edited and extended by _Klaus Brockhaus_ and _Robert G. Wilson v_, May 21 2007