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A213836 Rectangular array: (row n) = b**c, where b(h) = 4*h-3, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

%I A213836 #8 Jul 10 2012 19:30:13
%S A213836 1,7,2,22,13,3,50,37,19,4,95,78,52,25,5,161,140,106,67,31,6,252,227,
%T A213836 185,134,82,37,7,372,343,293,230,162,97,43,8,525,492,434,359,275,190,
%U A213836 112,49,9,715,678,612,525,425,320,218,127
%N A213836 Rectangular array:  (row n) = b**c, where b(h) = 4*h-3, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.
%C A213836 Principal diagonal: A213837.
%C A213836 Antidiagonal sums: A071238.
%C A213836 Row 1, (1,5,9,13,...)**(1,2,3,4,...): A002412.
%C A213836 Row 2, (1,5,9,13,...)**(2,3,4,5,...): (4*k^3 + 15*k^2 - 7*k)/6.
%C A213836 Row 3, (1,5,9,13,...)**(3,4,5,6,...): (4*k^3 + 27*k^2 - 13*k)/6.
%C A213836 For a guide to related arrays, see A212500.
%H A213836 Clark Kimberling, <a href="/A213836/b213836.txt">Antidiagonals n = 1..60, flattened</a>
%F A213836 T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
%F A213836 G.f. for row n: f(x)/g(x), where f(x) = x*(n + (2*n+1)*x - 3*(n-1)*x^2) and g(x) = (1-x)^4.
%e A213836 Northwest corner (the array is read by falling antidiagonals):
%e A213836 1...7....22...50....95
%e A213836 2...13...37...78....140
%e A213836 3...19...52...106...185
%e A213836 4...25...67...134...230
%e A213836 5...31...82...162...275
%e A213836 6...37...97...190...320
%t A213836 b[n_]:=4n-3;c[n_]:=n;
%t A213836 t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
%t A213836 TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
%t A213836 Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
%t A213836 r[n_]:=Table[t[n,k],{k,1,60}] (* A213836 *)
%t A213836 Table[t[n,n],{n,1,40}] (* A213837 *)
%t A213836 s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
%t A213836 Table[s[n],{n,1,50}] (* A071238 *)
%Y A213836 Cf. A212500.
%K A213836 nonn,tabl,easy
%O A213836 1,2
%A A213836 _Clark Kimberling_, Jul 04 2012