A213561 - OEIS

%I #9 Jul 12 2012 12:22:55

%S 1,7,3,27,18,6,77,61,34,10,182,157,109,55,15,378,342,267,171,81,21,

%T 714,665,557,407,247,112,28,1254,1190,1043,827,577,337,148,36,2079,

%U 1998,1806,1512,1152,777,441,189,45,3289,3189,2946,2562,2072,1532

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

%C Principal diagonal:  A213562

%C Antidiagonal sums:  A213563

%C Row 1,  (1,4,9,...)**(1,3,6,...):  A005585

%C Row 2,  (1,4,9,...)**(3,6,10,...):  (2*k^5 +25*k^4 + 120*k^3 + 155*k^2 + 58*k)/120

%C Row 3,  (1,4,9,...)**(6,10,15,...):  (2*k^5 +35*k^4 + 60*k^3 + 325*k^2 + 118*k)/120

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

%H Clark Kimberling, <a href="/A213561/b213561.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) - (n^2 - n - 2)*x - (n^2 + n - 2)*x^2 + n*(n - 1)*x^3 and g(x) = 2*(1 - x)^6.

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

%e 1....7.....27....77....182

%e 3....18....61....157...342

%e 6....34....109...267...557

%e 10...55....171...407...827

%e 15...81....247...577...1152

%e 21...112...337...777...1532

%t b[n_] := n^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}]  (* A213561 *)

%t d = Table[t[n, n], {n, 1, 40}] (* A213562 *)

%t s1 = Table[s[n], {n, 1, 50}] (* A213563 *)

%Y Cf. A213500.

%K nonn,tabl,easy

%O 1,2

%A _Clark Kimberling_, Jun 18 2012