A213833 - OEIS

%I #10 Feb 27 2018 10:27:25

%S 1,7,3,24,17,5,58,48,27,7,115,102,72,37,9,201,185,146,96,47,11,322,

%T 303,255,190,120,57,13,484,462,405,325,234,144,67,15,693,668,602,507,

%U 395,278,168,77,17,955,927,852,742,609,465

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

%C Principal diagonal: A103748.

%C Antidiagonal sums: A213834.

%C Row 1, (1,3,5,7,...)**(1,3,5,7,...): A081436.

%C Row 2, (1,3,5,7,...)**(3,5,7,9,...): A144640.

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

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

%H Clark Kimberling, <a href="/A213833/b213833.txt">Antidiagonals n = 1..12, 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) + (2*n+1)*x - (4*n-6)*x^2) and g(x) = (1-x)^4.

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

%e 1....7....24....58....115

%e 3....17...48....102...185

%e 5....27...72....146...255

%e 7....37...96....190...325

%e 9....47...120...234...395

%e 11...57...144...278...465

%t b[n_]:=3n-2;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}] (* A213833 *)

%t Table[t[n,n],{n,1,40}] (* A130748 *)

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

%t Table[s[n],{n,1,50}] (* A213834 *)

%Y Cf. A212500.

%K nonn,tabl,easy

%O 1,2

%A _Clark Kimberling_, Jul 04 2012