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A129819 Antidiagonal sums of triangular array T: T(j,k) = (k+1)/2 for odd k, T(j,k) = 0 for k = 0, T(j,k) = j+1-k/2 for even k > 0; 0 <= k <= j.

%I A129819 #24 Sep 19 2024 14:41:19
%S A129819 0,0,1,1,3,4,7,8,12,14,19,21,27,30,37,40,48,52,61,65,75,80,91,96,108,
%T A129819 114,127,133,147,154,169,176,192,200,217,225,243,252,271,280,300,310,
%U A129819 331,341,363,374,397,408,432,444,469,481,507,520,547,560,588,602,631
%N A129819 Antidiagonal sums of triangular array T: T(j,k) = (k+1)/2 for odd k, T(j,k) = 0 for k = 0, T(j,k) = j+1-k/2 for even k > 0; 0 <= k <= j.
%C A129819 Interleaving of A077043 and A006578.
%C A129819 First differences are in A124072.
%C A129819 If the values of the second, fourth, sixth, ... column are replaced by the corresponding negative values, the antidiagonal sums of the resulting triangular array are 0, 0, 1, 1, -1, -2, -1, -2, -6, -8, -7, -9, ... .
%C A129819 Row sums of triangle A168316 = (1, 1, 3, 4, 7, 8, 12, ...). - _Gary W. Adamson_, Nov 22 2009
%H A129819 G. C. Greubel, <a href="/A129819/b129819.txt">Table of n, a(n) for n = 0..1000</a>
%F A129819 a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-6) + a(n-7) for n > 6, with a(0) = 0, a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 3, a(5) = 4, a(6) = 7.
%F A129819 G.f.: x^2*(1+x^2+x^3)/((1-x)^3*(1+x)^2*(1+x^2)).
%F A129819 a(n) = (3/16)*(n+2)*(n+1) - (5/8)*(n+1) + 7/32 + (3/32)*(-1)^n + (1/16)*(n+1)*(-1)^n - (1/8)*cos(n*Pi/2) + (1/8)*sin(n*Pi/2). - _Richard Choulet_, Nov 27 2008
%e A129819 First seven rows of T are
%e A129819   0;
%e A129819   0, 1;
%e A129819   0, 1, 2;
%e A129819   0, 1, 3, 2;
%e A129819   0, 1, 4, 2, 3;
%e A129819   0, 1, 5, 2, 4, 3;
%e A129819   0, 1, 6, 2, 5, 3, 4;.
%t A129819 CoefficientList[Series[x^2*(1+x^2+x^3)/((1-x)*(1-x^2)*(1-x^4)), {x, 0, 70}], x] (* _G. C. Greubel_, Sep 19 2024 *)
%o A129819 (Magma) m:=59; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do for k:=2 to j do if k mod 2 eq 0 then M[j, k]:= k div 2; else M[j, k]:=j-(k div 2); end if; end for; end for; [ &+[ M[j-k+1, k]: k in [1..(j+1) div 2] ]: j in [1..m] ]; // _Klaus Brockhaus_, Jul 16 2007
%o A129819 (Magma)
%o A129819 A129819:= func< n | Floor(((n-1)*(3*n+1) +(2*n+5)*((n+1) mod 2))/16) >;
%o A129819 [A129819(n): n in [0..70]]; // _G. C. Greubel_, Sep 19 2024
%o A129819 (PARI) {vector(59, n, (n-2+n%2)*(n+n%2)/8+floor((n-2-n%2)^2/16))} \\ _Klaus Brockhaus_, Jul 16 2007
%o A129819 (SageMath)
%o A129819 def A129819(n): return ((n-1)*(3*n+1) + (2*n+5)*((n+1)%2))//16
%o A129819 [A129819(n) for n in range(71)] # _G. C. Greubel_, Sep 19 2024
%Y A129819 Cf. A006578, A077043, A124072, A168316.
%K A129819 nonn
%O A129819 0,5
%A A129819 _Paul Curtz_, May 20 2007
%E A129819 Edited and extended by _Klaus Brockhaus_, Jul 16 2007