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A207974 Triangle related to A152198.

%I A207974 #37 Mar 14 2017 20:47:33
%S A207974 1,1,1,1,2,1,1,3,1,1,1,4,2,2,1,1,5,2,4,1,1,1,6,3,6,3,2,1,1,7,3,9,3,5,
%T A207974 1,1,1,8,4,12,6,8,4,2,1,1,9,4,16,6,14,4,6,1,1,1,10,5,20,10,20,10,10,5,
%U A207974 2,1
%N A207974 Triangle related to A152198.
%C A207974 Row sums are A027383(n).
%C A207974 Diagonal sums are alternately A014739(n) and A001911(n+1).
%C A207974 The matrix inverse starts
%C A207974 1;
%C A207974 -1,1;
%C A207974 1,-2,1;
%C A207974 1,-1,-1,1;
%C A207974 -1,2,0,-2,1;
%C A207974 -1,1,2,-2,-1,1;
%C A207974 1,-2,-1,4,-1,-2,1;
%C A207974 1,-1,-3,3,3,-3,-1,1;
%C A207974 -1,2,2,-6,0,6,-2,-2,1;
%C A207974 -1,1,4,-4,-6,6,4,-4,-1,1;
%C A207974 1,-2,-3,8,2,-12,2,8,-3,-2,1;
%C A207974 apparently related to A158854. - _R. J. Mathar_, Apr 08 2013
%C A207974 From _Gheorghe Coserea_, Jun 11 2016: (Start)
%C A207974 T(n,k) is the number of terms of the sequence A057890 in the interval [2^n,2^(n+1)-1] having binary weight k+1.
%C A207974 T(n,k) = A007318(n,k) (mod 2) and the number of odd terms in row n of the triangle is 2^A000120(n).
%C A207974 (End)
%H A207974 Gheorghe Coserea, <a href="/A207974/b207974.txt">Rows n = 0..200, flattened</a>
%F A207974 T(n,k) = T(n-1,k-1) - (-1)^k*T(n-1,k), k>0 ; T(n,0) = 1.
%F A207974 T(2n,2k) = T(2n+1,2k) = binomial(n,k) = A007318(n,k).
%F A207974 T(2n+1,2k+1) = A110813(n,k).
%F A207974 T(2n+2,2k+1) = 2*A135278(n,k).
%F A207974 T(n,2k) + T(n,2k+1) = A152201(n,k).
%F A207974 T(n,2k) = A152198(n,k).
%F A207974 T(n+1,2k+1) = A152201(n,k).
%F A207974 T(n,k) = T(n-2,k-2) + T(n-2,k).
%F A207974 T(2n,n) = A128014(n+1).
%F A207974 T(n,k) = card {p, 2^n <= A057890(p) <= 2^(n+1)-1 and A000120(A057890(p)) = k+1}. - _Gheorghe Coserea_, Jun 09 2016
%F A207974 P_n(x) = Sum_{k=0..n} T(n,k)*x^k = ((2+x+(n mod 2)*x^2)*(1+x^2)^(n\2) - 2)/x. - _Gheorghe Coserea_, Mar 14 2017
%e A207974 Triangle begins :
%e A207974 n\k  [0] [1] [2] [3] [4] [5] [6] [7] [8] [9]
%e A207974 [0]  1;
%e A207974 [1]  1,  1;
%e A207974 [2]  1,  2,  1;
%e A207974 [3]  1,  3,  1,  1;
%e A207974 [4]  1,  4,  2,  2,  1;
%e A207974 [5]  1,  5,  2,  4,  1,  1;
%e A207974 [6]  1,  6,  3,  6,  3,  2,  1;
%e A207974 [7]  1,  7,  3,  9,  3,  5,  1,  1;
%e A207974 [8]  1,  8,  4,  12, 6,  8,  4,  2,  1;
%e A207974 [9]  1,  9,  4,  16, 6,  14, 4,  6,  1,  1;
%e A207974 [10] ...
%p A207974 A207974 := proc(n,k)
%p A207974     if k = 0 then
%p A207974         1;
%p A207974     elif k < 0 or k > n then
%p A207974         0 ;
%p A207974     else
%p A207974         procname(n-1,k-1)-(-1)^k*procname(n-1,k) ;
%p A207974     end if;
%p A207974 end proc: # _R. J. Mathar_, Apr 08 2013
%o A207974 (PARI)
%o A207974 seq(N) = {
%o A207974   my(t = vector(N+1, n, vector(n, k, k==1 || k == n)));
%o A207974   for(n = 2, N+1, for (k = 2, n-1,
%o A207974       t[n][k] = t[n-1][k-1] + (-1)^(k%2)*t[n-1][k]));
%o A207974   return(t);
%o A207974 };
%o A207974 concat(seq(10))  \\ _Gheorghe Coserea_, Jun 09 2016
%o A207974 (PARI)
%o A207974 P(n) = ((2+x+(n%2)*x^2) * (1+x^2)^(n\2) - 2)/x;
%o A207974 concat(vector(11, n, Vecrev(P(n-1)))) \\ _Gheorghe Coserea_, Mar 14 2017
%Y A207974 Cf. Columns : A000012, A000027, A004526, A002620, A008805, A006918, A058187
%Y A207974 Cf. Diagonals : A000012, A000034, A052938, A097362
%Y A207974 Cf. A007318, A110813, A135278, A152201
%Y A207974 Related to thickness: A000120, A027383, A057890, A274036.
%K A207974 easy,nonn,tabl
%O A207974 0,5
%A A207974 _Philippe Deléham_, Feb 22 2012