%I #10 Jul 11 2012 12:15:25

%S 2,13,5,42,28,8,98,78,43,11,190,164,114,58,14,327,295,230,150,73,17,

%T 518,480,400,296,186,88,20,772,728,633,505,362,222,103,23,1098,1048,

%U 938,786,610,428,258,118,26,1505,1449,1324

%N Rectangular array: (row n) = b**c, where b(h) = 3*h-2, c(h) = 3*n-4+3*h, n>=1, h>=1, and ** = convolution.

%C Principal diagonal: A213829.

%C Antidiagonal sums: A213830.

%C Row 1, (1,4,7,10,...)**(2,5,8,11,...): (3*k^2 + k)/2.

%C Row 2, (1,4,7,10,...)**(5,8,11,14,...): (3*k^3 + 9*k^2 + 4*k)/2.

%C Row 3, (1,4,7,10,...)**(8,11,14,17,...): (3*k^3 + 18*k^2 + 7*k)/2.

%C For a guide to related arrays, see A212500.

%H Clark Kimberling, <a href="/A213828/b213828.txt">Antidiagonals n=1..60, flattened</a>

%F T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

%F G.f. for row n: f(x)/g(x), where f(x) = x*((6*n-4) - (3*n-8)*x - (3*n-5)*x^2) and g(x) = (1-x)^4.

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

%e 2....13...42....98....190

%e 5....28...78....164...295

%e 8....43...114...230...400

%e 11...58...150...296...505

%e 14...73...186...362...610

%e 17...88...222...428...715

%t b[n_]:=3n-2;c[n_]:=3n-1;

%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}] (* A213828 *)

%t d=Table[t[n,n],{n,1,40}] (* A213829 *)

%t d/2 (* A005915 *)

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

%t Table[s[n],{n,1,50}] (* A213830 *)

%Y Cf. A212500

%K nonn,tabl,easy

%O 1,1

%A _Clark Kimberling_, Jul 04 2012