A213838 - OEIS

%I #11 Jul 11 2012 04:30:23

%S 1,8,3,29,20,5,72,59,32,7,145,128,89,44,9,256,235,184,119,56,11,413,

%T 388,325,240,149,68,13,624,595,520,415,296,179,80,15,897,864,777,652,

%U 505,352,209,92,17,1240,1203,1104,959,784

%N Rectangular array:  (row n) = b**c, where b(h) = 4*h-3, c(h) = 2*n-3+2*h, n>=1, h>=1, and ** = convolution.

%C Principal diagonal: A213839.

%C Antidiagonal sums: A213840.

%C Row 1, (1,5,9,13,...)**(1,3,5,7,...): A100178.

%C Row 2, (1,5,9,13,...)**(3,5,7,9,...): (4*k^3 + 9*k^2 - 4*k)/3.

%C Row 3, (1,5,9,13,...)**(5,7,9,11,...): (4*k^3 + 21*k^2 - 10*k)/3.

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

%H Clark Kimberling, <a href="/A213838/b213838.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*(2*n-1 + 4*n*x - (6*n-9)*x^2) and g(x) = (1-x)^4.

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

%e 1....8....29....72....145

%e 3....20...59....128...235

%e 5....32...89....184...325

%e 7....44...119...240...415

%e 9....56...149...296...505

%e 11...68...179...352...595

%t b[n_]:=4n-3; c[n_]:=2n-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}] (* A213838 *)

%t Table[t[n,n],{n,1,40}] (* A213839 *)

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

%t Table[s[n],{n,1,50}] (* A213840 *)

%Y Cf. A212500.

%K nonn,tabl,easy

%O 1,2

%A _Clark Kimberling_, Jul 05 2012