A176306 - OEIS

%I #15 Sep 08 2022 08:45:52

%S 1,1,1,1,0,1,1,13,13,1,1,25,38,25,1,1,-185,-160,-160,-185,1,1,-779,

%T -964,-952,-964,-779,1,1,7497,6718,6520,6520,6718,7497,1,1,45907,

%U 53404,52612,52402,52612,53404,45907,1,1,-524629,-478722,-471238,-472042,-472042,-471238,-478722,-524629,1

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

%C Row sums are {1, 2, 2, 28, 90, -688, -4436, 41472, 356250, -3893260, -41666708, ...}.

%H G. C. Greubel, <a href="/A176306/b176306.txt">Rows n = 0..100 of triangle, flattened</a>

%F T(n,k) = T(n,n-k).

%e Triangle begins as:

%e   1;

%e   1,     1;

%e   1,     0,     1;

%e   1,    13,    13,     1;

%e   1,    25,    38,    25,     1;

%e   1,  -185,  -160,  -160,  -185,     1;

%e   1,  -779,  -964,  -952,  -964,  -779,     1;

%e   1,  7497,  6718,  6520,  6520,  6718,  7497,     1;

%e   1, 45907, 53404, 52612, 52402, 52612, 53404, 45907, 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 A176306 := 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 0,8

%A _Roger L. Bagula_, Apr 14 2010