cp's OEIS Frontend

A213822 Rectangular array: (row n) = b**c, where b(h) = 3*h-1, c(h) = 3*n-4+3*h, n>=1, h>=1, and ** = convolution.

%I A213822 #10 Jul 12 2012 12:10:22
%S A213822 4,20,10,57,41,16,124,102,62,22,230,202,147,83,28,384,350,280,192,104,
%T A213822 34,595,555,470,358,237,125,40,872,826,726,590,436,282,146,46,1224,
%U A213822 1172,1057,897,710,514,327,167,52,1660,1602
%N A213822 Rectangular array:  (row n) = b**c, where b(h) = 3*h-1, c(h) = 3*n-4+3*h, n>=1, h>=1, and ** = convolution.
%C A213822 Principal diagonal: A213823.
%C A213822 Antidiagonal sums: A213824.
%C A213822 Row 1, (2,5,8,11,...)**(2,5,8,11,...): (3*k^3 + 3*k^2 + 2*k)/2.
%C A213822 Row 2, (2,5,8,11,...)**(5,8,11,14,...): (3*k^3 + 12*k^2 + 5*k)/2.
%C A213822 Row 3, (2,5,8,11,...)**(8,11,14,17,...): (3*k^3 + 21*k^2 + 8*k)/2.
%C A213822 For a guide to related arrays, see A212500.
%H A213822 Clark Kimberling, <a href="/A213822/b213822.txt">Antidiagonals n = 1..80, flattened</a>
%F A213822 T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
%F A213822 G.f. for row n: f(x)/g(x), where f(x) = x*((6*n-2) - (3*n-7)*x - (3*n-4)*x^2) and g(x) = (1-x)^4.
%e A213822 Northwest corner (the array is read by falling antidiagonals):
%e A213822 4....20....57....124...230
%e A213822 10...41....102...202...350
%e A213822 16...62....147...280...470
%e A213822 22...83....192...358...590
%e A213822 28...104...237...436...710
%t A213822 b[n_]:=3n-1;c[n_]:=3n-1;
%t A213822 t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
%t A213822 TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
%t A213822 Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
%t A213822 r[n_]:=Table[t[n,k],{k,1,60}] (* A213822 *)
%t A213822 Table[t[n,n],{n,1,40}] (* A213823 *)
%t A213822 s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
%t A213822 Table[s[n],{n,1,50}] (* A213824 *)
%Y A213822 Cf. A212500.
%K A213822 nonn,tabl,easy
%O A213822 1,1
%A A213822 _Clark Kimberling_, Jul 04 2012