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Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3*n-4+3*h, n>=1, h>=1, and ** = convolution.
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%I #13 Jul 16 2012 05:06:42

%S 2,9,5,24,18,8,50,42,27,11,90,80,60,36,14,147,135,110,78,45,17,224,

%T 210,180,140,96,54,20,324,308,273,225,170,114,63,23,450,432,392,336,

%U 270,200,132,72,26,605,585,540,476,399,315

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

%C Principal diagonal: A213820.

%C Antidiagonal sums: A153978.

%C Row 1, (1,2,3,4,...)**(2,5,8,11,...): A006002.

%C Row 2, (1,2,3,4,...)**(5,8,11,14,...): is it the sequence A212343?.

%C Row 3, (1,2,3,4,...)**(8,11,14,17,...): (k^3 + 8*k^2 + 7*k)/2.

%C For a guide to related arrays, see A212500.

%H Clark Kimberling, <a href="/A213819/b213819.txt">Antidiagonals n = 1..60, flattened</a>

%F T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

%F G.f. for row n: f(x)/g(x), where f(x) = x(3*n-1 - (3*n-4)*x) and g(x) = (1-x)^4.

%e Northwest corner (the array is read by falling antidiagonals):

%e 2....9....24....50....90....147

%e 5....18...42....80....135...210

%e 8....27...60....110...180...273

%e 11...36...78....140...225...336

%e 14...45...96....170...270...399

%e 17...54...114...200...315...462

%t b[n_]:=n;c[n_]:=3n-1;

%t t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]

%t TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]

%t Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]

%t r[n_]:=Table[t[n,k],{k,1,60}] (* A213819 *)

%t Table[t[n,n],{n,1,40}] (* A213820 *)

%t d/2 (* A002414 *)

%t s[n_]:=Sum[t[i,n+1-i],{i,1,n}]

%t Table[s[n],{n,1,50}] (* A153978 *)

%t s1/2 (* A001296 *)

%Y Cf. A212500

%K nonn,tabl,easy

%O 1,1

%A _Clark Kimberling_, Jul 04 2012