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A110555 Triangle of partial sums of alternating binomial coefficients: T(n, k) = Sum_{j=0..k} binomial(n, j)*(-1)^j, for n >= 0, 0 <= k <= n.

%I A110555 #42 Apr 13 2023 06:44:28
%S A110555 1,1,0,1,-1,0,1,-2,1,0,1,-3,3,-1,0,1,-4,6,-4,1,0,1,-5,10,-10,5,-1,0,1,
%T A110555 -6,15,-20,15,-6,1,0,1,-7,21,-35,35,-21,7,-1,0,1,-8,28,-56,70,-56,28,
%U A110555 -8,1,0,1,-9,36,-84,126,-126,84,-36,9,-1,0,1,-10,45,-120,210,-252,210,-120
%N A110555 Triangle of partial sums of alternating binomial coefficients: T(n, k) = Sum_{j=0..k} binomial(n, j)*(-1)^j, for n >= 0, 0 <= k <= n.
%H A110555 G. C. Greubel, <a href="/A110555/b110555.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A110555 Ângela Mestre and José Agapito, <a href="https://www.emis.de/journals/JIS/VOL22/Agapito/mestre8.html">A Family of Riordan Group Automorphisms</a>, J. Int. Seq., Vol. 22 (2019), Article 19.8.5.
%H A110555 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A110555 T(n, 0) = 1, T(n, n) = 0^n, T(n, k) = -T(n-1, k-1) + T(n-1, k), for 0 < k < n.
%F A110555 T(n, k) = binomial(n-1, k)*(-1)^k, 0 <= k < n, T(n, n) = 0^n.
%F A110555 T(n, n-k-1) = -T(n, k), for 0 < k < n.
%F A110555 T(n, k) = A071919(n, k)*(-1)^k and A071919(n, k) = abs(T(n, k)).
%F A110555 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [0, -1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Sep 05 2005
%F A110555 G.f.: (1 + x*y) / (1 + x*y - x). - _R. J. Mathar_, Aug 11 2015
%e A110555 Triangle T(n, k) starts:
%e A110555   [0] 1;
%e A110555   [1] 1,  0;
%e A110555   [2] 1, -1,  0;
%e A110555   [3] 1, -2,  1,   0;
%e A110555   [4] 1, -3,  3,  -1,  0;
%e A110555   [5] 1, -4,  6,  -4,  1,   0;
%e A110555   [6] 1, -5, 10, -10,  5,  -1,  0;
%e A110555   [7] 1, -6, 15, -20, 15,  -6,  1,  0;
%e A110555   [8] 1, -7, 21, -35, 35, -21,  7, -1,  0.
%p A110555 T := (n, k) -> (-1)^k * binomial(n-1, k):
%p A110555 seq(print(seq(T(n, k), k = 0..n)), n = 0..7); # _Peter Luschny_, Apr 13 2023
%t A110555 T[0, 0] := 1;  T[n_, n_] := 0; T[n_, k_] := (-1)^k*Binomial[n - 1, k]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* _G. C. Greubel_, Aug 31 2017 *)
%o A110555 (PARI) concat(1, for(n=1,10, for(k=0,n, print1(if(k != n, (-1)^k*binomial(n-1,k), 0), ", ")))) \\ _G. C. Greubel_, Aug 31 2017
%Y A110555 T(n,1)  = -n + 1 for n>0;
%Y A110555 T(n,2)  =  A000217(n-2) for n > 1;
%Y A110555 T(n,3)  = -A000292(n-4) for n > 2;
%Y A110555 T(n,4)  =  A000332(n-1) for n > 3;
%Y A110555 T(n,5)  = -A000389(n-1) for n > 5;
%Y A110555 T(n,6)  =  A000579(n-1) for n > 6;
%Y A110555 T(n,7)  = -A000580(n-1) for n > 7;
%Y A110555 T(n,8)  =  A000581(n-1) for n > 8;
%Y A110555 T(n,9)  = -A000582(n-1) for n > 9;
%Y A110555 T(n,10) =  A001287(n-1) for n > 10;
%Y A110555 T(n,11) = -A001288(n-1) for n > 11;
%Y A110555 T(n,12) =  A010965(n-1) for n > 12;
%Y A110555 T(n,13) = -A010966(n-1) for n > 13;
%Y A110555 T(n,14) =  A010967(n-1) for n > 14;
%Y A110555 T(n,15) = -A010968(n-1) for n > 15;
%Y A110555 T(n,16) =  A010969(n-1) for n > 16.
%Y A110555 Cf. A071919 (variant), A000007 (row sums), A110556 (central terms).
%Y A110555 Cf. A008949, A007318.
%K A110555 sign,easy,tabl
%O A110555 0,8
%A A110555 _Reinhard Zumkeller_, Jul 27 2005
%E A110555 Typo in name corrected by _Andrey Zabolotskiy_, Feb 22 2022
%E A110555 Offset corrected by _Peter Luschny_, Apr 13 2023