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A213773 Rectangular array: (row n) = b**c, where b(h) = 3*h-2, c(h) = 3*n-5+3*h, n>=1, h>=1, and ** = convolution.

%I A213773 #6 Jul 07 2012 08:37:45
%S A213773 1,8,4,30,23,7,76,66,38,10,155,142,102,53,13,276,260,208,138,68,16,
%T A213773 448,429,365,274,174,83,19,680,658,582,470,340,210,98,22,981,956,868,
%U A213773 735,575,406,246,113,25,1360,1332,1232,1078
%N A213773 Rectangular array:  (row n) = b**c, where b(h) = 3*h-2, c(h) = 3*n-5+3*h, n>=1, h>=1, and ** = convolution.
%C A213773 Principal diagonal: A213782
%C A213773 Antidiagonal sums: A214092
%C A213773 Row 1, (1,4,7,10,…)**(1,4,7,10,…): A100175
%C A213773 Row 2, (1,4,7,10,…)**(4,7,10,13,…): (3*k^3 + 6*k^2 - k)/2
%C A213773 Row 3, (1,4,7,10,…)**(7,10,13,16,…): (3*k^3 + 15*k^2 - 4*k)/2
%C A213773 For a guide to related arrays, see A212500.
%H A213773 Clark Kimberling, <a href="/A213773/b213773.txt">Table of n, a(n) for n = 1..1034</a>
%F A213773 T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
%F A213773 G.f. for row n: f(x)/g(x), where f(x) = x(3*n-2 + (3*n+1)*x - (6*n-10)*x^2) and g(x) = (1-x)^4.
%e A213773 Northwest corner (the array is read by falling antidiagonals):
%e A213773 1....8....30....76....155...276
%e A213773 4....23...66....142...260...429
%e A213773 7....38...102...208...365...582
%e A213773 10...53...138...274...470...735
%e A213773 13...68...174...340...575...888
%t A213773 b[n_]:=3n-2;c[n_]:=3n-2;
%t A213773 t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
%t A213773 TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
%t A213773 Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
%t A213773 r[n_]:=Table[t[n,k],{k,1,60}] (* A213773 *)
%t A213773 Table[t[n,n],{n,1,40}] (* A214092 *)
%t A213773 s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
%t A213773 Table[s[n],{n,1,50}] (* A213818 *)
%Y A213773 Cf. A213500.
%K A213773 nonn,tabl,easy
%O A213773 1,2
%A A213773 _Clark Kimberling_, Jul 04 2012