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A176306 Triangle T(n,k) = 1 - A176304(k) - A176304(n-k) + A176304(n), read by rows.

%I A176306 #15 Sep 08 2022 08:45:52
%S A176306 1,1,1,1,0,1,1,13,13,1,1,25,38,25,1,1,-185,-160,-160,-185,1,1,-779,
%T A176306 -964,-952,-964,-779,1,1,7497,6718,6520,6520,6718,7497,1,1,45907,
%U A176306 53404,52612,52402,52612,53404,45907,1,1,-524629,-478722,-471238,-472042,-472042,-471238,-478722,-524629,1
%N A176306 Triangle T(n,k) = 1 - A176304(k) - A176304(n-k) + A176304(n), read by rows.
%C A176306 Row sums are {1, 2, 2, 28, 90, -688, -4436, 41472, 356250, -3893260, -41666708, ...}.
%H A176306 G. C. Greubel, <a href="/A176306/b176306.txt">Rows n = 0..100 of triangle, flattened</a>
%F A176306 T(n,k) = T(n,n-k).
%e A176306 Triangle begins as:
%e A176306   1;
%e A176306   1,     1;
%e A176306   1,     0,     1;
%e A176306   1,    13,    13,     1;
%e A176306   1,    25,    38,    25,     1;
%e A176306   1,  -185,  -160,  -160,  -185,     1;
%e A176306   1,  -779,  -964,  -952,  -964,  -779,     1;
%e A176306   1,  7497,  6718,  6520,  6520,  6718,  7497,     1;
%e A176306   1, 45907, 53404, 52612, 52402, 52612, 53404, 45907, 1;
%p A176306 A176304 := proc(n)
%p A176306     if n = 0 then
%p A176306         0;
%p A176306     else
%p A176306         (-1)^n*n*procname(n-1)-1 ;
%p A176306     end if;
%p A176306 end proc:
%p A176306 A176306 := proc(n,m)
%p A176306     1-A176304(m)-A176304(n-m)+A176304(n) ;
%p A176306 end proc: # _R. J. Mathar_, May 04 2013
%t A176306 b[n_]:= b[n] = If[n==0, 0, (-1)^n*n*b[n-1] -1];
%t A176306 T[n_, k_]:= T[n, k] = 1 - (b[k] +b[n-k] -b[n]);
%t A176306 Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten
%o A176306 (PARI) b(n) = if(n==0, 0, (-1)^n*n*b(n-1) -1);
%o A176306 T(n,k) = 1 + b(n) - b(k) - b(n-k); \\ _G. C. Greubel_, Nov 26 2019
%o A176306 (Magma)
%o A176306 function b(n)
%o A176306   if n eq 0 then return 0;
%o A176306   else return (-1)^n*n*b(n-1) -1;
%o A176306   end if; return b; end function;
%o A176306 [1+b(n)-b(k)-b(n-k): k in [0..n], n in [1..10]]; // _G. C. Greubel_, Nov 26 2019
%o A176306 (Sage)
%o A176306 @CachedFunction
%o A176306 def b(n):
%o A176306     if (n==0): return 0
%o A176306     else: return (-1)^n*n*b(n-1) -1
%o A176306 [[1+b(n)-b(k)-b(n-k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 26 2019
%K A176306 sign,tabl
%O A176306 0,8
%A A176306 _Roger L. Bagula_, Apr 14 2010