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A176307 Triangle T(n,k) = 1 + A176304(k) + A176304(n-k) - A176304(n), read by rows.

%I A176307 #15 Sep 08 2022 08:45:52
%S A176307 1,1,1,1,2,1,1,-11,-11,1,1,-23,-36,-23,1,1,187,162,162,187,1,1,781,
%T A176307 966,954,966,781,1,1,-7495,-6716,-6518,-6518,-6716,-7495,1,1,-45905,
%U A176307 -53402,-52610,-52400,-52610,-53402,-45905,1,1,524631,478724,471240,472044,472044,471240,478724,524631,1
%N A176307 Triangle T(n,k) = 1 + A176304(k) + A176304(n-k) - A176304(n), read by rows.
%C A176307 Row sums are {1, 2, 4, -20, -80, 700, 4450, -41456, -356232, 3893280, 41666730, ...}.
%H A176307 G. C. Greubel, <a href="/A176307/b176307.txt">Rows n = 0..100 of triangle, flattened</a>
%F A176307 T(n,k) = T(n,n-k) = 2 - A176306(n,k).
%e A176307 Triangle begins as:
%e A176307   1;
%e A176307   1,     1;
%e A176307   1,     2,     1;
%e A176307   1,   -11,   -11,      1;
%e A176307   1,   -23,   -36,    -23,      1;
%e A176307   1,   187,   162,    162,    187,      1;
%e A176307   1,   781,   966,    954,    966,    781,      1;
%e A176307   1, -7495, -6716,  -6518,  -6518,  -6716,  -7495,    1;
%p A176307 A176304 := proc(n)
%p A176307     if n = 0 then
%p A176307         0;
%p A176307     else
%p A176307         (-1)^n*n*procname(n-1)-1 ;
%p A176307     end if;
%p A176307 end proc:
%p A176307 A176307 := proc(n,m)
%p A176307     1+A176304(m)+A176304(n-m)-A176304(n) ;
%p A176307 end proc: # _R. J. Mathar_, May 04 2013
%t A176307 b[n_]:= b[n]= If[n==0, 0, (-1)^n*n*b[n-1] -1];
%t A176307 T[n_, k_]:= T[n, k] = 1 + b[k] + b[n-k] - b[n];
%t A176307 Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten
%o A176307 (PARI) b(n) = if(n==0, 0, (-1)^n*n*b(n-1) -1);
%o A176307 T(n,k) = 1 - b(n) + b(k) + b(n-k); \\ _G. C. Greubel_, Nov 26 2019
%o A176307 (Magma)
%o A176307 function b(n)
%o A176307   if n eq 0 then return 0;
%o A176307   else return (-1)^n*n*b(n-1) -1;
%o A176307   end if; return b; end function;
%o A176307 [1-b(n)+b(k)+b(n-k): k in [0..n], n in [1..10]]; // _G. C. Greubel_, Nov 26 2019
%o A176307 (Sage)
%o A176307 @CachedFunction
%o A176307 def b(n):
%o A176307     if (n==0): return 0
%o A176307     else: return (-1)^n*n*b(n-1) -1
%o A176307 [[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 A176307 sign,tabl
%O A176307 0,5
%A A176307 _Roger L. Bagula_, Apr 14 2010