A213821 - OEIS

%I #6 Jul 06 2012 09:16:00

%S 2,9,4,24,16,6,50,39,23,8,90,76,54,30,10,147,130,102,69,37,12,224,204,

%T 170,128,84,44,14,324,301,261,210,154,99,51,16,450,424,378,318,250,

%U 180,114,58,18,605,576,524,455,375,290,206

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

%C Principal diagonal: A033431.

%C Antidiagonal sums: A176060.

%C Row 1, (2,5,8,11,…)**(1,2,3,4,…): A006002.

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

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

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

%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*(2*n - (n-2)*x - (n-1)*x^2) and g(x) = (1-x)^4.

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

%e 2….9….24…50….90

%e 4….16…39…76…130

%e 6….23…54…102…170

%e 8….30…69…128…210

%e 10…37…84…154…250

%e 12…44…99…180…290

%t b[n_]:=3n-1;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}] (* A213821 *)

%t Table[t[n,n],{n,1,40}] (* A033431 *)

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

%t Table[s[n],{n,1,50}] (* A176060 *)

%Y Cf. A212500

%K nonn,tabl,easy

%O 1,1

%A _Clark Kimberling_, Jul 04 2012