cp's OEIS Frontend

A213821 Rectangular array: (row n) = b**c, where b(h) = 3*h-1, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

%I A213821 #6 Jul 06 2012 09:16:00
%S A213821 2,9,4,24,16,6,50,39,23,8,90,76,54,30,10,147,130,102,69,37,12,224,204,
%T A213821 170,128,84,44,14,324,301,261,210,154,99,51,16,450,424,378,318,250,
%U A213821 180,114,58,18,605,576,524,455,375,290,206
%N A213821 Rectangular array:  (row n) = b**c, where b(h) = 3*h-1, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.
%C A213821 Principal diagonal: A033431.
%C A213821 Antidiagonal sums: A176060.
%C A213821 Row 1, (2,5,8,11,…)**(1,2,3,4,…): A006002.
%C A213821 Row 2, (2,5,8,11,…)**(2,3,4,5,…): (k^3 + 5*k^2 + 2*k)/2.
%C A213821 Row 3, (1,2,3,4,…)**(8,11,14,17,…): (k^3 + 8*k^2 + 3*k)/2.
%C A213821 For a guide to related arrays, see A212500.
%F A213821 T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
%F A213821 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 A213821 Northwest corner (the array is read by falling antidiagonals):
%e A213821 2….9….24…50….90
%e A213821 4….16…39…76…130
%e A213821 6….23…54…102…170
%e A213821 8….30…69…128…210
%e A213821 10…37…84…154…250
%e A213821 12…44…99…180…290
%t A213821 b[n_]:=3n-1;c[n_]:=n;
%t A213821 t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
%t A213821 TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
%t A213821 Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
%t A213821 r[n_]:=Table[t[n,k],{k,1,60}] (* A213821 *)
%t A213821 Table[t[n,n],{n,1,40}] (* A033431 *)
%t A213821 s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
%t A213821 Table[s[n],{n,1,50}] (* A176060 *)
%Y A213821 Cf. A212500
%K A213821 nonn,tabl,easy
%O A213821 1,1
%A A213821 _Clark Kimberling_, Jul 04 2012