A213841 - OEIS

%I #9 Jul 13 2012 11:41:43

%S 1,8,5,29,24,9,72,65,40,13,145,136,101,56,17,256,245,200,137,72,21,

%T 413,400,345,264,173,88,25,624,609,544,445,328,209,104,29,897,880,805,

%U 688,545,392,245,120,33,1240,1221,1136,1001

%N Rectangular array:  (row n) = b**c, where b(h) = 2*h-1, c(h) = 4*n-7+4*h, n>=1, h>=1, and ** = convolution.

%C Principal diagonal: A213842.

%C Antidiagonal sums: A213843.

%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 + 2*k)/3.

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

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

%H Clark Kimberling, <a href="/A213841/b213841.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*(4*n-3 + 4*x - (4*n-7)*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 5....24...65....136...245

%e 9....40...101...200...345

%e 13...56...137...264...445

%e 17...72...173...328...545

%e 21...88...209...392...645

%t b[n_]:=2n-1;c[n_]:=4n-3;

%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}] (* A213841 *)

%t Table[t[n,n],{n,1,40}] (* A213842 *)

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

%t Table[s[n],{n,1,50}] (* A213843 *)

%Y Cf. A212500.

%K nonn,tabl,easy

%O 1,2

%A _Clark Kimberling_, Jul 05 2012