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A213853 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = binomial(2*n-4+2*h,n-2+h), n>=1, h>=1, and ** = convolution.

%I A213853 #15 Jun 13 2025 01:09:08
%S A213853 1,4,2,13,10,6,42,38,32,20,141,136,128,110,70,492,486,476,452,392,252,
%T A213853 1767,1760,1748,1718,1638,1428,924,6474,6466,6452,6416,6316,6036,5280,
%U A213853 3432,24051,24042,24026,23984,23864,23514,22506,19734,12870
%N A213853 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = binomial(2*n-4+2*h,n-2+h), n>=1, h>=1, and ** = convolution.
%C A213853 Row 1, (1,2,3,4,5,...)**(1,2,6,20,70,...):
%C A213853 Row 2, (1,2,3,4,5,...)**(2,6,20,70,252,...):
%C A213853 Row 3, (1,2,3,4,5,...)**(6,20,70,252,...):
%C A213853 For a guide to related arrays, see A213500.
%H A213853 Clark Kimberling, <a href="/A213853/b213853.txt">Antidiagonals n = 1..60, flattened</a>
%e A213853 Northwest corner (the array is read by falling antidiagonals):
%e A213853   1    4     13     42     141
%e A213853   2    10    38     136    486
%e A213853   6    32    128    476    1748
%e A213853   20   110   452    1718   6416
%e A213853   70   392   1638   6316   23864
%t A213853 b[n_]:=n;c[n_]:=Binomial[2n-2,n-1]
%t A213853 t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
%t A213853 TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
%t A213853 Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
%t A213853 r[n_]:=Table[t[n,k],{k,1,20}] (* A213853 *)
%Y A213853 Cf. A213500.
%K A213853 nonn,tabl,easy
%O A213853 1,2
%A A213853 _Clark Kimberling_, Jul 05 2012