A176340 - OEIS

%I #13 Sep 08 2022 08:45:52

%S 1,1,1,1,-8,1,1,190,190,1,1,-14822,-14624,-14822,1,1,3557278,3542464,

%T 3542464,3557278,1,1,-2582583830,-2579026544,-2579041556,-2579026544,

%U -2582583830,1,1,5640363084718,5637780500896,5637784057984,5637784057984,5637780500896,5640363084718,1

%N Triangle T(n,k) = 1 - A176338(k) - A176338(n-k) + A176338(n) read by rows.

%C Row sums are: {1, 2, -6, 382, -44266, 14199486, -12902262302, 33831855287198,

%C -258898313695820850, 5823405140242006622494, -386839522966544578870468774, ...}.

%H G. C. Greubel, <a href="/A176340/b176340.txt">Rows n = 0..25 of triangle, flattened</a>

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

%e Triangle starts as:

%e   1;

%e   1,       1;

%e   1,      -8,       1;

%e   1,     190,     190,       1;

%e   1,  -14822,  -14624,  -14822,       1;

%e   1, 3557278, 3542464, 3542464, 3557278, 1;

%t b[n_, q_]:= b[n, q]= If[n==0, 0, (1-q^n)*b[n-1, q] +1];

%t T[n_,k_,q_]:= 1 + b[n,q] -b[n-k,q] - b[k,q];

%t Table[T[n,k,3], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Dec 07 2019 *)

%o (PARI) b(n,q) = if(n==0, 0, 1 + (1-q^n)*b(n-1,q) );

%o T(n,k,q) = 1 + b(n,q) - b(n-k,q) - b(k,q);

%o for(n=0,10, for(k=0,n, print1(T(n,k,3), ", "))) \\ _G. C. Greubel_, Dec 07 2019

%o (Magma)

%o function b(n,q)

%o   if n eq 0 then return 0;

%o   else return 1 - (q^n-1)*b(n-1,q);

%o   end if; return b; end function;

%o function T(n,k,q) return 1 + b(n,q) - b(n-k,q) - b(k,q); end function;

%o [ T(n,k,3) : k in [0..n], n in [0..10]]; // _G. C. Greubel_, Dec 07 2019

%o (Sage)

%o @CachedFunction

%o def b(n, q):

%o     if (n==0): return 0

%o     else: return 1 - (q^n-1)*b(n-1,q)

%o def T(n,k,q): return 1 + b(n,q) - b(n-k,q) - b(k,q)

%o [[T(n,k,3) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Dec 07 2019

%o (GAP)

%o b:= function(n,q)

%o     if n=0 then return 0;

%o     else return 1 - (q^n-1)*b(n-1,q);

%o     fi; end;

%o T:= function(n,k,q) return 1 + b(n,q) - b(n-k,q) - b(k,q); end;

%o Flat(List([0..10], n-> List([0..n], k-> T(n,k,3) ))); # _G. C. Greubel_, Dec 07 2019

%Y Cf. A176337, A176338, A176339.

%K sign,tabl

%O 0,5

%A _Roger L. Bagula_, Apr 15 2010

%E Edited by _G. C. Greubel_, Dec 07 2019