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%I #43 Feb 26 2020 13:47:53
%S 1,1,-1,1,0,1,1,1,1,-1,1,2,2,0,1,1,3,4,2,1,-1,1,4,7,6,3,0,1,1,5,11,13,
%T 9,3,1,-1,1,6,16,24,22,12,4,0,1,1,7,22,40,46,34,16,4,1,-1,1,8,29,62,
%U 86,80,50,20,5,0,1,1,9,37,91,148,166,130,70,25,5,1,-1,1,10,46,128,239,314,296,200,95,30,6,0,1,1,11,56,174,367
%N Triangle read by rows: T(n,0)=1, T(n,n)=(-1)^n, T(n+1,k)=T(n,k-1)+T(n,k) for 0 < k < n.
%C Sum_{k=0..n} T(n,k) = A078008(n);
%C Sum_{k=0..n} abs(T(n,k) = A052953(n-1) for n > 0;
%C T(n,1) = n - 2 for n > 1;
%C T(n,2) = A000124(n-3) for n > 2;
%C T(n,3) = A003600(n-4) for n > 4;
%C T(n,n-6) = A001753(n-6) for n > 6;
%C T(n,n-5) = A001752(n-5) for n > 5;
%C T(n,n-4) = A002623(n-4) for n > 4;
%C T(n,n-3) = A002620(n-1) for n > 3;
%C T(n,n-2) = A008619(n-2) for n > 2;
%C T(n,n-1) = n mod 2 for n > 0;
%C T(2*n,n) = A072547(n+1).
%C Sum_{k=0..n} T(n,k)*x^k = A232015(n), A078008(n), A000012(n), A040000(n), A001045(n+2), A140725(n+1) for x = 2, 1, 0, -1, -2, -3 respectively. - _Philippe Deléham_, Nov 17 2013, Nov 19 2013
%C (1,a^n) Pascal triangle with a = -1. - _Philippe Deléham_, Dec 27 2013
%C T(n,k) = A112465(n,n-k). - _Reinhard Zumkeller_, Jan 03 2014
%H Reinhard Zumkeller, <a href="/A108561/b108561.txt">Rows n = 0..125 of triangle, flattened</a>
%H C. Merino, S. D. Noble, M. Ramirez-Ibanez, R. Villarroel-Flores, <a href="https://doi.org/10.1016/j.ejc.2012.04.002">On the structure of the h-vector of a paving matroid</a>, Eur. J. Comb. 33, No. 8, 1787-1799 (2012).
%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F G.f.: (1-y*x)/(1-x-(y+y^2)*x). - _Philippe Deléham_, Nov 17 2013
%F T(n,k) = T(n-1,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0)=T(1,0)=1, T(1,1)=-1, T(n,k)=0 if k < 0 or if k > n. - _Philippe Deléham_, Nov 17 2013
%F From _Peter Bala_, Feb 18 2018: (Start)
%F T(n,k) = Sum_{i = 0..k} binomial(n,i)*(-2)^(k-i), 0 <= k <= n.
%F The n-th row polynomial is the n-th degree Taylor polynomial of the rational function (1 + x)^n/(1 + 2*x) about 0. For example, for n = 4, (1 + x)^4/(1 + 2*x) = 1 + 2*x + 2*x^2 + x^4 + O(x^5). (End)
%e From _Philippe Deléham_, Nov 17 2013: (Start)
%e Triangle begins:
%e 1;
%e 1, -1;
%e 1, 0, 1;
%e 1, 1, 1, -1;
%e 1, 2, 2, 0, 1;
%e 1, 3, 4, 2, 1, -1;
%e 1, 4, 7, 6, 3, 0, 1; (End)
%p A108561 := (n, k) -> add(binomial(n, i)*(-2)^(k-i), i = 0..k):
%p seq(seq(A108561(n,k), k = 0..n), n = 0..12); # _Peter Bala_, Feb 18 2018
%t Clear[t]; t[n_, 0] = 1; t[n_, n_] := t[n, n] = (-1)^Mod[n, 2]; t[n_, k_] := t[n, k] = t[n-1, k] + t[n-1, k-1]; Table[t[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Mar 06 2013 *)
%o (Haskell)
%o a108561 n k = a108561_tabl !! n !! k
%o a108561_row n = a108561_tabl !! n
%o a108561_tabl = map reverse a112465_tabl
%o -- _Reinhard Zumkeller_, Jan 03 2014
%o (Sage)
%o def A108561_row(n):
%o @cached_function
%o def prec(n, k):
%o if k==n: return 1
%o if k==0: return 0
%o return -prec(n-1,k-1)-sum(prec(n,k+i-1) for i in (2..n-k+1))
%o return [(-1)^k*prec(n, k) for k in (1..n-1)]+[(-1)^(n+1)]
%o for n in (1..12): print(A108561_row(n)) # _Peter Luschny_, Mar 16 2016
%o (GAP) Flat(List([0..13],n->List([0..n],k->Sum([0..k],i->Binomial(n,i)*(-2)^(k-i))))); # _Muniru A Asiru_, Feb 19 2018
%Y Cf. A007318 (a=1), A008949(a=2), A164844(a=10).
%Y Similar to the triangles A035317, A059259, A080242, A112555.
%Y Cf. A228196
%Y Cf. A072547 (central terms).
%K sign,tabl
%O 0,12
%A _Reinhard Zumkeller_, Jun 10 2005
%E Definition corrected by _Philippe Deléham_, Dec 26 2013
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