%I #10 May 14 2017 12:21:56

%S 1,5,2,15,9,3,35,25,13,4,70,55,35,17,5,126,105,75,45,21,6,210,182,140,

%T 95,55,25,7,330,294,238,175,115,65,29,8,495,450,378,294,210,135,75,33,

%U 9,715,660,570,462,350,245,155,85,37,10,1001,935,825,690,546

%N Rectangular array: (row n) = b**c, where b(h) = h*(h+1)/2, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

%C Principal diagonal: A002418

%C Antidiagonal sums: A005585

%C row 1, (1,3,6,...)**(1,2,3,...): A000332

%C row 2, (1,3,6,...)**(2,3,4,...): A005582

%C row 3, (1,3,6,...)**(3,4,5,...): A095661

%C row 4, (1,3,6,...)**(4,5,6,...): A095667

%C For a guide to related arrays, see A213500.

%F T(n,k) = 5*T(n,k-1) - 10*T(n,k-2) + 10*T(n,k-3) - 5*T(n,k-4) + T(n,k-5).

%F G.f. for row n: f(x)/g(x), where f(x) = n-(n-1)*x and g(x) = (1 - x)^5.

%e Northwest corner (the array is read by falling antidiagonals):

%e 1....5....15...35....70....126

%e 2....9....25...55....105...182

%e 3....13...35...75....140...238

%e 4....17...45...95....175...294

%e 5....21...55...115...210...350

%t b[n_] := n (n + 1)/2; c[n_] := n

%t t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]

%t TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]

%t Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]

%t r[n_] := Table[t[n, k], {k, 1, 60}] (* A213550 *)

%t d = Table[t[n, n], {n, 1, 40}] (* A002418 *)

%t s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]

%t s1 = Table[s[n], {n, 1, 50}] (* A005585 *)

%Y Cf. A213500, A213548.

%K nonn,tabl,easy

%O 1,2

%A _Clark Kimberling_, Jun 16 2012