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A292781 Triangle read by rows: T(n,k) = T(n-k,k-1) with T(0,0) = 1 and T(n,0) = -1/n * Sum_{k=1..A003056(n)} (-1)^k * (2*k+1) * (n+1-A060544(k+1)) * T(n,k).

%I A292781 #26 Oct 23 2018 19:17:29
%S A292781 1,-24,1,252,-24,-1472,252,1,4830,-1472,-24,-6048,4830,252,-16744,
%T A292781 -6048,-1472,1,84480,-16744,4830,-24,-113643,84480,-6048,252,-115920,
%U A292781 -113643,-16744,-1472,534612,-115920,84480,4830,1,-370944,534612,-113643,-6048,-24
%N A292781 Triangle read by rows: T(n,k) = T(n-k,k-1) with T(0,0) = 1 and T(n,0) = -1/n * Sum_{k=1..A003056(n)} (-1)^k * (2*k+1) * (n+1-A060544(k+1)) * T(n,k).
%H A292781 Seiichi Manyama, <a href="/A292781/b292781.txt">Rows n = 0..481, flattened</a>
%e A292781 First few rows are:
%e A292781         1;
%e A292781       -24,       1;
%e A292781       252,     -24;
%e A292781     -1472,     252,      1;
%e A292781      4830,   -1472,    -24;
%e A292781     -6048,    4830,    252;
%e A292781    -16744,   -6048,  -1472,     1;
%e A292781     84480,  -16744,   4830,   -24;
%e A292781   -113643,   84480,  -6048,   252;
%e A292781   -115920, -113643, -16744, -1472;
%e A292781    534612, -115920,  84480,  4830, 1.
%e A292781 -----------------------------------------
%e A292781 n=5
%e A292781 T(5,1) = T(4,0) = 4830,  T(5,2) = T(3,1) = 252.
%e A292781 T(5,0) = -1/5 * Sum_{k=1..2} (-1)^k * (2*k+1) * (5+1-A060544(k+1)) * T(n,k) = -1/5 * ((-3)*(-4)*4830 + 5*(-22)*252) = -6048.
%e A292781 n=6
%e A292781 T(6,1) = T(5,0) = -6048,  T(6,2) = T(4,1) = -1472, T(6,3) = T(3,2) = 1.
%e A292781 T(6,0) = -1/6 * Sum_{k=1..3} (-1)^k * (2*k+1) * (6+1-A060544(k+1)) * T(n,k) = -1/6 * ((-3)*(-3)*(-6048) + 5*(-21)*(-1472) - 7*(-48)*1) = -16744.
%o A292781 (Ruby)
%o A292781 def A292781(n)
%o A292781   ary = [[1]]
%o A292781   (1..n).each{|i|
%o A292781     m = ((Math.sqrt(1 + 8 * i) - 1) / 2).to_i
%o A292781     a = (1..m).map{|j| ary[i - j][j - 1]}
%o A292781     ary << [-(1..m).inject(0){|s, j| s + (-1) ** (j % 2) * (2 * j + 1) * (i - 9 * j * (j + 1) / 2) * a[j - 1]} / i] + a
%o A292781   }
%o A292781   ary.flatten
%o A292781 end
%o A292781 p A292781(20)
%Y A292781 All columns are A000594.
%Y A292781 Cf. A003056, A060544.
%K A292781 sign,tabf
%O A292781 0,2
%A A292781 _Seiichi Manyama_, Sep 23 2017