A213551 - OEIS

%I #9 Jul 15 2012 04:09:54

%S 1,6,3,21,15,6,56,46,28,10,126,111,81,45,15,252,231,186,126,66,21,462,

%T 434,371,281,181,91,28,792,756,672,546,396,246,120,36,1287,1242,1134,

%U 966,756,531,321,153,45,2002,1947,1812,1596,1316,1001,686,406

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

%C Principal diagonal:  A213552

%C Antidiagonal sums:  A051923

%C Row 1,  (1,3,6,...)**(1,3,6,...): A000389

%C Row 2,  (1,3,6,...)**(3,6,10,...): (k^5 + 15*k^4 + 85*k^3 + 165*k^2 + 94*k)/120

%C Row 3,  (1,3,6,...)**(6,10,15,...): (k^5 + 20*k^4 + 155*k^3 + 340*k^2 + 204*k)/120

%C For a guide to related arrays, see A213500.

%H Clark Kimberling, <a href="/A213551/b213551.txt">Antidiagonals n = 1..60, flattened</a>

%F T(n,k) = 6*T(n,k-1) - 15*T(n,k-2) + 20*T(n,k-3) - 15*T(n,k-4) + 6*T(n,k-5) - T(n,k-6).

%F G.f. for row n:  f(x)/g(x), where f(x) = n*(n+1) - 2*((n-1)^2)*x + 2*(n-1)*x^2 and g(x) = 2*(1 - x)^2.

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

%e 1....6....21....56....126....252

%e 3....15...46....111...231....434

%e 6....28...81....186...371....672

%e 10...45...126...281...546....966

%e 15...66...181...396...756....1316

%e 21...91...246...531...1001...1722

%t b[n_] := n (n + 1)/2; c[n_] := n (n + 1)/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}]  (* A213551 *)

%t d = Table[t[n, n], {n, 1, 40}] (* A213552 *)

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

%t s1 = Table[s[n], {n, 1, 50}] (* A051923 *)

%Y Cf. A213500.

%K nonn,tabl,easy

%O 1,2

%A _Clark Kimberling_, Jun 17 2012