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a(n) = Sum_{k=0..floor(n/2)} A323833(n,k) if A323833 is read as a triangle.
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%I #16 Mar 05 2021 02:31:36

%S 0,1,1,-5,-8,61,130,-1385,-3680,50521,160816,-2702765,-10026368,

%T 199360981,844583440,-19391512145,-92369507840,2404879675441,

%U 12722897618176,-370371188237525,-2154662195222528,69348874393137901,440001333689382400,-15514534163557086905,-106615331831035289600,4087072509293123892361

%N a(n) = Sum_{k=0..floor(n/2)} A323833(n,k) if A323833 is read as a triangle.

%C Because A323833(n,n/2) = 0 for n even (if A323833 is read as a triangle), we also have a(n) = Sum_{k=0..ceiling((n-1)/2)} A323833(n,k).

%F a(n) = Sum_{k=0..floor(n/2)} A323833(n-k,k) if A323833 is read as a square array (by upwards antidiagonals).

%F a(2*n+1) = -A028296(n+1).

%F a(n) = Sum_{k=0..floor(n/2)} Sum_{i=0..n-k} binomial(n-k,i) * (-1)^(k+i) * A163747(k+i).

%e a(3) = -2 - 3 = -5.

%e a(4) = -5 - 3 = -8.

%e a(5) = 16 + 21 + 24 = 61.

%e a(6) = 61 + 45 + 24 = 130.

%e a(7) = -272 - 333 - 378 - 402 = -1385.

%o (PARI) {b(n) = local(v=[1], t); if( n<0, 0, for(k=2, n+2, t=0; v = vector(k, i, if( i>1, t+= v[k+1-i]))); v[2])}; \\ _Michael Somos_'s PARI program for A000111

%o c(n) = if(n==0, 0, (-1)^(n+floor(n/2))*b(n))

%o a(n) = sum(k=0, floor(n/2), sum(i=0, n-k, binomial(n-k,i)*(-1)^(k+i)*c(k+i)))

%Y Cf. A000111, A028296, A163747, A323833.

%K sign

%O 0,4

%A _Petros Hadjicostas_, Mar 04 2021