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A117411 Skew triangle associated to the Euler numbers.

%I A117411 #22 Sep 07 2022 08:13:42
%S A117411 1,0,1,0,-4,1,0,0,-12,1,0,0,16,-24,1,0,0,0,80,-40,1,0,0,0,-64,240,-60,
%T A117411 1,0,0,0,0,-448,560,-84,1,0,0,0,0,256,-1792,1120,-112,1,0,0,0,0,0,
%U A117411 2304,-5376,2016,-144,1,0,0,0,0,0,-1024,11520,-13440,3360,-180,1,0,0,0,0,0,0,-11264,42240,-29568,5280,-220,1
%N A117411 Skew triangle associated to the Euler numbers.
%C A117411 Inverse is A117414. Row sums of the inverse are the Euler numbers A000364.
%C A117411 Triangle, read by rows, given by [0,-4,4,0,0,0,0,0,0,0,...] DELTA [1,0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Nov 01 2009
%H A117411 G. C. Greubel, <a href="/A117411/b117411.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A117411 Sum_{k=0..n} T(n, k) = A006495(n).
%F A117411 Sum_{k=0..floor(n/2)} T(n-k, k) = A117413(n).
%F A117411 T(n, k) = (-4)^(n-k)*Sum_{j=0..n-k} C(n,k-j)*C(j,n-k).
%F A117411 G.f.: (1-x*y)/(1-2x*y+x^2*y(y+4)). - _Paul Barry_, Mar 14 2006
%F A117411 T(n, k) = (-4)^(n-k)*A098158(n,k). - _Philippe Deléham_, Nov 01 2009
%F A117411 T(n, k) = 2*T(n-1,k-1) - 4*T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(1,1) = 1, T(1,0) = 0, T(n,k) = 0 if k > n or if k < 0. - _Philippe Deléham_, Oct 31 2013
%F A117411 From _G. C. Greubel_, Sep 07 2022: (Start)
%F A117411 T(n, n) = 1.
%F A117411 T(n, n-1) = -4*A000217(n-1), n >= 1.
%F A117411 T(n, n-2) = (-4)^2 * A000332(n), n >= 2.
%F A117411 T(n, n-3) = (-4)^3 * A000579(n), n >= 3.
%F A117411 T(n, n-4) = (-4)^4 * A000581(n), n >= 4.
%F A117411 T(2*n, n) = A262710(n). (End)
%e A117411 Triangle begins
%e A117411   1;
%e A117411   0,  1;
%e A117411   0, -4,   1;
%e A117411   0,  0, -12,   1;
%e A117411   0,  0,  16, -24,    1;
%e A117411   0,  0,   0,  80,  -40,     1;
%e A117411   0,  0,   0, -64,  240,   -60,      1;
%e A117411   0,  0,   0,   0, -448,   560,    -84,      1;
%e A117411   0,  0,   0,   0,  256, -1792,   1120,   -112,      1;
%e A117411   0,  0,   0,   0,    0,  2304,  -5376,   2016,   -144,      1;
%e A117411   0,  0,   0,   0,    0, -1024,  11520, -13440,   3360,   -180,    1;
%e A117411   0,  0,   0,   0,    0,     0, -11264,  42240, -29568,   5280, -220,    1;
%e A117411   0,  0,   0,   0,    0,     0,   4096, -67584, 126720, -59136, 7920, -264, 1;
%t A117411 T[n_,k_]:= T[n,k]= (-4)^(n-k)*Sum[Binomial[n, k-j]*Binomial[j, n-k], {j,0,n-k}];
%t A117411 Table[T[n,k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Sep 07 2022 *)
%o A117411 (Magma)
%o A117411 A117411:= func< n,k | (-4)^(n-k)*(&+[Binomial(n,k-j)*Binomial(j,n-k): j in [0..n-k]]) >;
%o A117411 [A117411(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Sep 07 2022
%o A117411 (SageMath)
%o A117411 def A117411(n,k): return (-4)^(n-k)*sum(binomial(n,k-j)*binomial(j,n-k) for j in (0..n-k))
%o A117411 flatten([[A117411(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Sep 07 2022
%Y A117411 Cf. A000364, A006495 (row sums), A098158, A117413, A117414.
%Y A117411 Cf. A000217, A000332, A000579, A000581, A262710.
%K A117411 easy,sign,tabl
%O A117411 0,5
%A A117411 _Paul Barry_, Mar 13 2006