A213835 - OEIS

%I #11 May 25 2018 04:23:45

%S 1,7,5,22,19,9,50,46,31,13,95,90,70,43,17,161,155,130,94,55,21,252,

%T 245,215,170,118,67,25,372,364,329,275,210,142,79,29,525,516,476,413,

%U 335,250,166,91,33,715,705,660,588,497,395

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

%C Principal diagonal: A172078.

%C Antidiagonal sums: A051797.

%C Row 1, (1,2,3,4,5,...)**(1,5,9,13,...): A002412.

%C Row 2, (1,2,3,4,5,...)**(5,9,13,17,...): (4*k^3 + 15*k^2 - 11*k)/6.

%C Row 3, (1,2,3,4,5,...)**(9,13,17,21,...): (4*k^3 + 27*k^2 - 23*k)/6

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

%H Clark Kimberling, <a href="/A213835/b213835.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*n-7)*x) and g(x) = (1-x)^4.

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

%e 1....7....22....50....95

%e 5....19...46....90....155

%e 9....31...70....130...215

%e 13...43...94....170...275

%e 17...55...118...210...335

%e 21...67...142...250...395

%t b[n_]:=n;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}] (* A213835 *)

%t Table[t[n,n],{n,1,40}] (* A172078 *)

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

%t Table[s[n],{n,1,50}] (* A051797 *)

%Y Cf. A212500.

%Y Cf. A304659 (first lower diagonal).

%K nonn,tabl,easy

%O 1,2

%A _Clark Kimberling_, Jul 04 2012