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A176305 Triangle T(n,k) = 1 -A002627(k) -A002627(n-k) +A002627(n), read by rows.

%I A176305 #11 Sep 08 2022 08:45:52
%S A176305 1,1,1,1,2,1,1,7,7,1,1,31,36,31,1,1,165,194,194,165,1,1,1031,1194,
%T A176305 1218,1194,1031,1,1,7423,8452,8610,8610,8452,7423,1,1,60621,68042,
%U A176305 69066,69200,69066,68042,60621,1,1,554249,614868,622284,623284,623284,622284,614868,554249,1
%N A176305 Triangle T(n,k) = 1 -A002627(k) -A002627(n-k) +A002627(n), read by rows.
%C A176305 Row sums are {1, 2, 4, 16, 100, 720, 5670, 48972, 464660, 4829372, 54711782, ...}.
%C A176305 Row sums s(n) appear to obey (2-n)*s(n) +(n+2)*(n-1)*s(n-2) -n*(2*n-1)*s(n-2) +n*(n-2)*s(n-3)=0. - _R. J. Mathar_, May 04 2013
%H A176305 G. C. Greubel, <a href="/A176305/b176305.txt">Rows n = 0..100 of triangle, flattened</a>
%F A176305 T(n,k) = T(n,n-k).
%e A176305 Triangle begins as:
%e A176305   1;
%e A176305   1,      1;
%e A176305   1,      2,      1;
%e A176305   1,      7,      7,      1;
%e A176305   1,     31,     36,     31,      1;
%e A176305   1,    165,    194,    194,    165,      1;
%e A176305   1,   1031,   1194,   1218,   1194,   1031,      1;
%e A176305   1,   7423,   8452,   8610,   8610,   8452,   7423,      1;
%e A176305   1,  60621,  68042,  69066,  69200,  69066,  68042,  60621,      1;
%e A176305   1, 554249, 614868, 622284, 623284, 623284, 622284, 614868, 554249, 1;
%p A176305 T:= proc(n, k) option remember;
%p A176305       if k=0 or k=n then 1
%p A176305     else 1 +floor(n!*(exp(1)-1)) -floor(k!*(exp(1)-1)) -floor((n-k)!*(exp(1)-1))
%p A176305       fi; end:
%p A176305 seq(seq(T(n, k), k=0..n), n=0..10); # _G. C. Greubel_, Nov 26 2019
%t A176305 (* First program *)
%t A176305 a[n_]:= a[n] = If[n==0,0,n*a[n-1] +1];
%t A176305 T[n_, k_]:= T[n, k] = 1 -(a[k] +a[n-k] -a[n]);
%t A176305 Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten
%t A176305 (* Second program *)
%t A176305 T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, 1 +Floor[n!*(E-1)] -Floor[k!*(E-1)] - Floor[(n-k)!*(E-1)]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 26 2019 *)
%o A176305 (PARI) T(n,k) = if(k==0 || k==n, 1, 1 +floor(n!*(exp(1)-1)) -floor(k!*(exp(1)-1)) -floor((n-k)!*(exp(1)-1)) ); \\ _G. C. Greubel_, Nov 26 2019
%o A176305 (Magma)b:= func< n | Factorial(n)*(Exp(1)-1)>;
%o A176305 function T(n,k)
%o A176305   if k eq 0 or k eq n then return 1;
%o A176305   else return 1 +Floor(b(n)) -Floor(b(k)) -Floor(b(n-k));
%o A176305   end if; return T; end function;
%o A176305 [T(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Nov 26 2019
%o A176305 (Sage)
%o A176305 @CachedFunction
%o A176305 def b(n): return factorial(n)*(exp(1)-1);
%o A176305 def T(n, k):
%o A176305     if (k==0 or k==n): return 1
%o A176305     else: return 1 +floor(b(n)) -floor(b(k)) -floor(b(n-k))
%o A176305 [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 26 2019
%K A176305 nonn,tabl
%O A176305 0,5
%A A176305 _Roger L. Bagula_, Apr 14 2010