A213831 - OEIS

%I #8 Jul 11 2012 05:03:39

%S 1,7,4,24,19,7,58,51,31,10,115,106,78,43,13,201,190,154,105,55,16,322,

%T 309,265,202,132,67,19,484,469,417,340,250,159,79,22,693,676,616,525,

%U 415,298,186,91,25,955,936,868,763,633

%N Rectangular array:  (row n) = b**c, where b(h) = 2*h-1, c(h) = 3*n-5+3*h, n>=1, h>=1, and ** = convolution.

%C Principal diagonal: A213832.

%C Antidiagonal sums: A212560.

%C row 1, (1,3,5,7,...)**(1,4,7,10,...): A081436.

%C Row 2, (1,3,5,7,...)**(4,7,10,13,...): A162254.

%C Row 3, (1,3,5,7,...)**(7,10,13,16,...): (2*k^3 + 11*k^2 + k)/2.

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

%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-2) + 3*x - (3*n-5)*x^2) and g(x) = (1-x)^4.

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

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

%e 4....19...51....106...190

%e 7....31...78....154...265

%e 10...43...105...202...340

%e 13...55...132...250...415

%t b[n_]:=2n-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}] (* A213831 *)

%t Table[t[n,n],{n,1,40}] (* A213832 *)

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

%t Table[s[n],{n,1,50}] (* A212560 *)

%Y Cf. A212500

%K nonn,tabl,easy

%O 1,2

%A _Clark Kimberling_, Jul 04 2012