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A338796 Triangle T read by rows: T(n, k) is the k-th row sum of the symmetric Toeplitz matrix M(n) whose first row consists of a single zero followed by successive positive integers repeated (A004526).

%I A338796 #45 Dec 22 2024 03:25:35
%S A338796 0,1,1,2,2,2,4,3,3,4,6,5,4,5,6,9,7,6,6,7,9,12,10,8,8,8,10,12,16,13,11,
%T A338796 10,10,11,13,16,20,17,14,13,12,13,14,17,20,25,21,18,16,15,15,16,18,21,
%U A338796 25,30,26,22,20,18,18,18,20,22,26,30,36,31,27,24,22,21,21,22,24,27,31,36
%N A338796 Triangle T read by rows: T(n, k) is the k-th row sum of the symmetric Toeplitz matrix M(n) whose first row consists of a single zero followed by successive positive integers repeated (A004526).
%F A338796 O.g.f.: (2*x^3*y^2 + y^2*(1 + y) + x^2*(y - 3*y^2) - x*(-1 + 2*y + y^2))/((1 - x)^3*(1 + x) *(1 - y)^3*(1 + y)).
%F A338796 E.g.f.: exp(-x-y)*(exp(2*x) + 2*exp(2*(x+y))*(x*(3 + x) - 2*x*y + 2*y^2 - 1))/8.
%F A338796 T(n, k) = ((-1)^k + (-1)^(n-k+1) + 4*k^2 + 4*n + 2*n^2 - 4*k*(n + 1))/8.
%F A338796 T(n, 1) = T(n, n) = A002620(n).
%F A338796 T(n, 2) = A033638(n-1).
%F A338796 T(n, 3) = A290743(n-2).
%F A338796 Sum_{k=1..n} T(n, k) = A212964(n+1).
%e A338796 n\k| 1 2 3 4 5 6
%e A338796 ---+------------
%e A338796 1  | 0
%e A338796 2  | 1 1
%e A338796 3  | 2 2 2
%e A338796 4  | 4 3 3 4
%e A338796 5  | 6 5 4 5 6
%e A338796 6  | 9 7 6 6 7 9
%e A338796 ...
%e A338796 For n = 4 the matrix M(4) is
%e A338796         0 1 1 2
%e A338796         1 0 1 1
%e A338796         1 1 0 1
%e A338796         2 1 1 0
%e A338796 and therefore T(4, 1) = 4, T(4, 2) = 3, T(4, 3) = 3 and T(4, 4) = 4.
%t A338796 T[n_,k_]:=((-1)^k+(-1)^(n-k+1)+4k^2+4n+2n^2-4k(n+1))/8; Flatten[Table[T[n,k],{n,12},{k,n}]] (* or *)
%t A338796 r[n_]:=Table[SeriesCoefficient[(2x^3y^2+y^2(1+y)+x^2(y-3y^2)-x(-1+2y+y^2))/((1-x)^3(1+x)(1-y)^3(1+y)),{x,0,i},{y,0,j}],{i,n,n},{j, n}]; Flatten[Array[r,12]] (* or *)
%t A338796 r[n_]:=Table[SeriesCoefficient[1/8 E^(-x-y)(-1+E^(2 x)+2 E^(2 (x+y))(x (3+x)-2 x y+2 y^2)),{x, 0, i},{y, 0, j}]i!j!,{i, n, n},{j, n}]; Flatten[Array[r, 12]]
%o A338796 (PARI) tm(n) = {my(m = matrix(n, n, i, j, if (i==1, j\2, if (j==1, i\2)))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m; }
%o A338796 T(n, k) = my(m = tm(n)); sum(i=1, n, m[i, k]);
%o A338796 matrix(10, 10, n, k, if (n>=k, T(n,k), 0)) \\ _Michel Marcus_, Nov 12 2020
%Y A338796 Cf. A004526.
%Y A338796 Cf. A002620, A033638, A212964, A290743.
%Y A338796 Cf. A002378 (conjectured determinant of M(2n+1)), A083392 (conjectured determinant of M(n+1)), A332566 (permanent of M(n)), A333119 (k-th super- and subdiagonal sums of the matrix M(n)), A338456 (hafnian of M(n)).
%K A338796 nonn,tabl
%O A338796 1,4
%A A338796 _Stefano Spezia_, Nov 12 2020