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A213831 Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = 3*n-5+3*h, n>=1, h>=1, and ** = convolution.

%I A213831 #8 Jul 11 2012 05:03:39
%S A213831 1,7,4,24,19,7,58,51,31,10,115,106,78,43,13,201,190,154,105,55,16,322,
%T A213831 309,265,202,132,67,19,484,469,417,340,250,159,79,22,693,676,616,525,
%U A213831 415,298,186,91,25,955,936,868,763,633
%N A213831 Rectangular array:  (row n) = b**c, where b(h) = 2*h-1, c(h) = 3*n-5+3*h, n>=1, h>=1, and ** = convolution.
%C A213831 Principal diagonal: A213832.
%C A213831 Antidiagonal sums: A212560.
%C A213831 row 1, (1,3,5,7,...)**(1,4,7,10,...): A081436.
%C A213831 Row 2, (1,3,5,7,...)**(4,7,10,13,...): A162254.
%C A213831 Row 3, (1,3,5,7,...)**(7,10,13,16,...): (2*k^3 + 11*k^2 + k)/2.
%C A213831 For a guide to related arrays, see A212500.
%F A213831 T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
%F A213831 G.f. for row n: f(x)/g(x), where f(x) = x*((3*n-2) + 3*x - (3*n-5)*x^2) and g(x) = (1-x)^4.
%F A213831 Northwest corner (the array is read by falling antidiagonals):
%e A213831 1....7....24....58....115
%e A213831 4....19...51....106...190
%e A213831 7....31...78....154...265
%e A213831 10...43...105...202...340
%e A213831 13...55...132...250...415
%t A213831 b[n_]:=2n-1;c[n_]:=3n-2;
%t A213831 t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
%t A213831 TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
%t A213831 Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
%t A213831 r[n_]:=Table[t[n,k],{k,1,60}] (* A213831 *)
%t A213831 Table[t[n,n],{n,1,40}] (* A213832 *)
%t A213831 s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
%t A213831 Table[s[n],{n,1,50}] (* A212560 *)
%Y A213831 Cf. A212500
%K A213831 nonn,tabl,easy
%O A213831 1,2
%A A213831 _Clark Kimberling_, Jul 04 2012