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