cp's OEIS Frontend

A117440 A cyclically signed version of Pascal's triangle.

%I A117440 #18 Sep 12 2024 03:03:40
%S A117440 1,1,1,-1,2,1,-1,-3,3,1,1,-4,-6,4,1,1,5,-10,-10,5,1,-1,6,15,-20,-15,6,
%T A117440 1,-1,-7,21,35,-35,-21,7,1,1,-8,-28,56,70,-56,-28,8,1,1,9,-36,-84,126,
%U A117440 126,-84,-36,9,1,-1,10,45,-120,-210,252,210,-120,-45,10,1
%N A117440 A cyclically signed version of Pascal's triangle.
%H A117440 G. C. Greubel, <a href="/A117440/b117440.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A117440 Column k has e.g.f.: (x^k/k!)*(cos(x) + sin(x)).
%F A117440 T(n, k) = binomial(n,k)*(cos(Pi*(n-k)/2) + sin(Pi*(n-k)/2)).
%F A117440 Sum_{k=0..n} T(n, k) = A009545(n+1).
%F A117440 Sum_{k=0..floor(n/2)} T(n-k, k) = A117441(n) (upward diagonal sums).
%F A117440 G.f.: (1 + x - x*y)/(1 - 2*x*y + x^2*(1 + y^2)). - _Stefano Spezia_, Mar 10 2024
%e A117440 Triangle begins:
%e A117440    1;
%e A117440    1,  1;
%e A117440   -1,  2,   1;
%e A117440   -1, -3,   3,   1;
%e A117440    1, -4,  -6,   4,   1;
%e A117440    1,  5, -10, -10,   5,   1;
%e A117440   -1,  6,  15, -20, -15,   6, 1;
%e A117440   -1, -7,  21,  35, -35, -21, 7, 1;
%t A117440 Table[Binomial[n, k]*(Cos[Pi*(n-k)/2] +Sin[Pi*(n-k)/2]), {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 01 2021 *)
%o A117440 (Sage) flatten([[binomial(n,k)*( cos(pi*(n-k)/2) + sin(pi*(n-k)/2) ) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 01 2021
%Y A117440 Cf. A007318, A009545 (row sums), A117441 (diagonal sums), A117442 (inverse).
%K A117440 easy,sign,tabl
%O A117440 0,5
%A A117440 _Paul Barry_, Mar 16 2006