A213822 - OEIS

%I #10 Jul 12 2012 12:10:22

%S 4,20,10,57,41,16,124,102,62,22,230,202,147,83,28,384,350,280,192,104,

%T 34,595,555,470,358,237,125,40,872,826,726,590,436,282,146,46,1224,

%U 1172,1057,897,710,514,327,167,52,1660,1602

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

%C Principal diagonal: A213823.

%C Antidiagonal sums: A213824.

%C Row 1, (2,5,8,11,...)**(2,5,8,11,...): (3*k^3 + 3*k^2 + 2*k)/2.

%C Row 2, (2,5,8,11,...)**(5,8,11,14,...): (3*k^3 + 12*k^2 + 5*k)/2.

%C Row 3, (2,5,8,11,...)**(8,11,14,17,...): (3*k^3 + 21*k^2 + 8*k)/2.

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

%H Clark Kimberling, <a href="/A213822/b213822.txt">Antidiagonals n = 1..80, 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*((6*n-2) - (3*n-7)*x - (3*n-4)*x^2) and g(x) = (1-x)^4.

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

%e 4....20....57....124...230

%e 10...41....102...202...350

%e 16...62....147...280...470

%e 22...83....192...358...590

%e 28...104...237...436...710

%t b[n_]:=3n-1;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}] (* A213822 *)

%t Table[t[n,n],{n,1,40}] (* A213823 *)

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

%t Table[s[n],{n,1,50}] (* A213824 *)

%Y Cf. A212500.

%K nonn,tabl,easy

%O 1,1

%A _Clark Kimberling_, Jul 04 2012