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A213561 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.

%I A213561 #9 Jul 12 2012 12:22:55
%S A213561 1,7,3,27,18,6,77,61,34,10,182,157,109,55,15,378,342,267,171,81,21,
%T A213561 714,665,557,407,247,112,28,1254,1190,1043,827,577,337,148,36,2079,
%U A213561 1998,1806,1512,1152,777,441,189,45,3289,3189,2946,2562,2072,1532
%N A213561 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 A213561 Principal diagonal:  A213562
%C A213561 Antidiagonal sums:  A213563
%C A213561 Row 1,  (1,4,9,...)**(1,3,6,...):  A005585
%C A213561 Row 2,  (1,4,9,...)**(3,6,10,...):  (2*k^5 +25*k^4 + 120*k^3 + 155*k^2 + 58*k)/120
%C A213561 Row 3,  (1,4,9,...)**(6,10,15,...):  (2*k^5 +35*k^4 + 60*k^3 + 325*k^2 + 118*k)/120
%C A213561 For a guide to related arrays, see A213500.
%H A213561 Clark Kimberling, <a href="/A213561/b213561.txt">Antidiagonals n = 1..60, flattened</a>
%F A213561 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 A213561 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 A213561 Northwest corner (the array is read by falling antidiagonals):
%e A213561 1....7.....27....77....182
%e A213561 3....18....61....157...342
%e A213561 6....34....109...267...557
%e A213561 10...55....171...407...827
%e A213561 15...81....247...577...1152
%e A213561 21...112...337...777...1532
%t A213561 b[n_] := n^2; c[n_] := n (n + 1)/2
%t A213561 t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
%t A213561 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213561 Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
%t A213561 r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213561 *)
%t A213561 d = Table[t[n, n], {n, 1, 40}] (* A213562 *)
%t A213561 s1 = Table[s[n], {n, 1, 50}] (* A213563 *)
%Y A213561 Cf. A213500.
%K A213561 nonn,tabl,easy
%O A213561 1,2
%A A213561 _Clark Kimberling_, Jun 18 2012