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A159855 Riordan array ((1-2*x-x^2)/(1-x), x/(1-x)).

%I A159855 #25 Sep 08 2022 08:45:44
%S A159855 1,-1,1,-2,0,1,-2,-2,1,1,-2,-4,-1,2,1,-2,-6,-5,1,3,1,-2,-8,-11,-4,4,4,
%T A159855 1,-2,-10,-19,-15,0,8,5,1,-2,-12,-29,-34,-15,8,13,6,1,-2,-14,-41,-63,
%U A159855 -49,-7,21,19,7,1,-2,-16,-55,-104,-112,-56,14,40,26,8,1
%N A159855 Riordan array ((1-2*x-x^2)/(1-x), x/(1-x)).
%C A159855 The matrix inverse starts
%C A159855    1;
%C A159855    1,  1;
%C A159855    2,  0,   1;
%C A159855    2,  2,  -1,   1;
%C A159855    4,  0,   3,  -2,   1;
%C A159855    4,  4,  -3,   5,  -3,    1;
%C A159855    8,  0,   7,  -8,   8,   -4,   1;
%C A159855    8,  8,  -7,  15, -16,   12,  -5,   1;
%C A159855   16,  0,  15, -22,  31,  -28,  17,  -6,  1;
%C A159855   16, 16, -15,  37, -53,   59, -45,  23, -7,  1;
%C A159855   32,  0,  31, -52,  90, -112, 104, -68, 30, -8, 1;
%C A159855 - _R. J. Mathar_, Mar 29 2013
%H A159855 Muniru A Asiru, <a href="/A159855/b159855.txt">Table of n, a(n) for n = 0..5151</a>
%F A159855 T(n,k) = C(n,n-k) - 2*C(n-1,n-k-1) - C(n-2,n-k-2), where C(n,k) = n!/(k!*(n-k)!) for 0 <= k <= n, otherwise 0. - _Peter Bala_, Mar 20 2018
%e A159855 Triangle begins:
%e A159855    1;
%e A159855   -1,  1;
%e A159855   -2,  0,  1;
%e A159855   -2, -2,  1,  1;
%e A159855   -2, -4, -1,  2,  1;
%e A159855   -2, -6, -5,  1,  3,  1;
%p A159855 C := proc (n, k) if 0 <= k and k <= n then factorial(n)/(factorial(k)*factorial(n-k)) else 0 end if end proc:
%p A159855 for n from 0 to 10 do
%p A159855   seq(C(n, n-k)-2*C(n-1, n-k-1)-C(n-2, n-k-2), k = 0..n)
%p A159855 end do; # _Peter Bala_, Mar 20 2018
%t A159855 (* The function RiordanArray is defined in A256893. *)
%t A159855 RiordanArray[(1 - 2 # - #^2)/(1 - #)&, #/(1 - #)&, 11] // Flatten (* _Jean-François Alcover_, Jul 16 2019 *)
%o A159855 (Sage) # uses[riordan_array from A256893]
%o A159855 riordan_array((1-2*x-x^2)/(1-x), x/(1-x), 8) # _Peter Luschny_, Mar 21 2018
%o A159855 (GAP) Flat(List([0..12],n->List([0..n],k->Binomial(n,n-k)-2*Binomial(n-1,n-k-1)-Binomial(n-2,n-k-2)))); # _Muniru A Asiru_, Mar 22 2018
%o A159855 (Magma) /* As triangle */ [[Binomial(n, n-k)-2*Binomial(n-1, n-k-1)-Binomial(n-2, n-k-2): k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Mar 22 2018
%Y A159855 Cf. A159854.
%K A159855 easy,sign,tabl
%O A159855 0,4
%A A159855 _Philippe Deléham_, Apr 24 2009
%E A159855 Two data values in row 10 corrected by _Peter Bala_, Mar 20 2018