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A213825 Rectangular array:  (row n) = b**c, where b(h) = 3*h-1, c(h) = 3*n-5+3*h, n>=1, h>=1, and ** = convolution. 4

%I

%S 2,13,8,42,34,14,98,87,55,20,190,176,132,76,26,327,310,254,177,97,32,

%T 518,498,430,332,222,118,38,772,749,669,550,410,267,139,44,1098,1072,

%U 980,840,670,488,312,160,50,1505,1476,1372

%N Rectangular array: (row n) = b**c, where b(h) = 3*h-1, c(h) = 3*n-5+3*h, n>=1, h>=1, and ** = convolution.

%C Principal diagonal: A213826

%C Antidiagonal sums: A213827

%C Row 1, (2,5,8,13,...)**(1,4,7,10,13,...): (3*k^2 + k)/2

%C Row 2, (2,5,8,13,...)**(4,7,10,13,...): (3*k^3 + 9*k^2 - 2*k)/2

%C Row 3, (2,5,8,13,...)**(7,10,13,16,...): (3*k^3 + 18*k^2 - 5*k)/2

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

%H Clark Kimberling, <a href="/A213825/b213825.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*((3*n-1) + (3*n+2)*x - (6*n-8)*x^2) and g(x) = (1-x)^4.

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

%e 2....13....42....98....190

%e 8....34....87....176...310

%e 14...55....132...254...430

%e 20...76....177...332...550

%e 26...97....222...410...670

%e 32...118...267...488...790

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

%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}] (* A213825 *)

%t d=Table[t[n,n],{n,1,40}] (* A213826 *)

%t d/2 (* A024215 *)

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

%t s1=Table[s[n],{n,1,50}] (* A213827 *)

%Y Cf. A212500

%K nonn,tabl,easy

%O 1,1

%A _Clark Kimberling_, Jul 04 2012

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Last modified May 7 06:57 EDT 2021. Contains 343636 sequences. (Running on oeis4.)