cp's OEIS Frontend

A213828 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.

%I A213828 #10 Jul 11 2012 12:15:25
%S A213828 2,13,5,42,28,8,98,78,43,11,190,164,114,58,14,327,295,230,150,73,17,
%T A213828 518,480,400,296,186,88,20,772,728,633,505,362,222,103,23,1098,1048,
%U A213828 938,786,610,428,258,118,26,1505,1449,1324
%N A213828 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 A213828 Principal diagonal: A213829.
%C A213828 Antidiagonal sums: A213830.
%C A213828 Row 1, (1,4,7,10,...)**(2,5,8,11,...): (3*k^2 + k)/2.
%C A213828 Row 2, (1,4,7,10,...)**(5,8,11,14,...): (3*k^3 + 9*k^2 + 4*k)/2.
%C A213828 Row 3, (1,4,7,10,...)**(8,11,14,17,...): (3*k^3 + 18*k^2 + 7*k)/2.
%C A213828 For a guide to related arrays, see A212500.
%H A213828 Clark Kimberling, <a href="/A213828/b213828.txt">Antidiagonals n=1..60, flattened</a>
%F A213828 T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
%F A213828 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 A213828 Northwest corner (the array is read by falling antidiagonals):
%e A213828 2....13...42....98....190
%e A213828 5....28...78....164...295
%e A213828 8....43...114...230...400
%e A213828 11...58...150...296...505
%e A213828 14...73...186...362...610
%e A213828 17...88...222...428...715
%t A213828 b[n_]:=3n-2;c[n_]:=3n-1;
%t A213828 t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
%t A213828 TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
%t A213828 Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
%t A213828 r[n_]:=Table[t[n,k],{k,1,60}] (* A213828 *)
%t A213828 d=Table[t[n,n],{n,1,40}] (* A213829 *)
%t A213828 d/2 (* A005915 *)
%t A213828 s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
%t A213828 Table[s[n],{n,1,50}] (* A213830 *)
%Y A213828 Cf. A212500
%K A213828 nonn,tabl,easy
%O A213828 1,1
%A A213828 _Clark Kimberling_, Jul 04 2012