login
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).
2

%I #26 Oct 23 2018 19:17:29

%S 1,-24,1,252,-24,-1472,252,1,4830,-1472,-24,-6048,4830,252,-16744,

%T -6048,-1472,1,84480,-16744,4830,-24,-113643,84480,-6048,252,-115920,

%U -113643,-16744,-1472,534612,-115920,84480,4830,1,-370944,534612,-113643,-6048,-24

%N 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 Seiichi Manyama, <a href="/A292781/b292781.txt">Rows n = 0..481, flattened</a>

%e First few rows are:

%e 1;

%e -24, 1;

%e 252, -24;

%e -1472, 252, 1;

%e 4830, -1472, -24;

%e -6048, 4830, 252;

%e -16744, -6048, -1472, 1;

%e 84480, -16744, 4830, -24;

%e -113643, 84480, -6048, 252;

%e -115920, -113643, -16744, -1472;

%e 534612, -115920, 84480, 4830, 1.

%e -----------------------------------------

%e n=5

%e T(5,1) = T(4,0) = 4830, T(5,2) = T(3,1) = 252.

%e 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 n=6

%e T(6,1) = T(5,0) = -6048, T(6,2) = T(4,1) = -1472, T(6,3) = T(3,2) = 1.

%e 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 (Ruby)

%o def A292781(n)

%o ary = [[1]]

%o (1..n).each{|i|

%o m = ((Math.sqrt(1 + 8 * i) - 1) / 2).to_i

%o a = (1..m).map{|j| ary[i - j][j - 1]}

%o 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 }

%o ary.flatten

%o end

%o p A292781(20)

%Y All columns are A000594.

%Y Cf. A003056, A060544.

%K sign,tabf

%O 0,2

%A _Seiichi Manyama_, Sep 23 2017