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A212891 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution.

%I A212891 #12 Jun 21 2012 12:17:47
%S A212891 1,6,4,20,17,9,50,46,34,16,105,100,84,57,25,196,190,170,134,86,36,336,
%T A212891 329,305,260,196,121,49,540,532,504,450,370,270,162,64,825,816,784,
%U A212891 721,625,500,356,209,81,1210,1200,1164,1092,980,830,650,454,262
%N A212891 Rectangular array:  (row n) = b**c, where b(h) = h, c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution.
%C A212891 Principal diagonal:  A213436
%C A212891 Antidiagonal sums:  A024166
%C A212891 row 1,  (1,2,3,...)**(1,4,9,...):  A002415(k+1)
%C A212891 row 2,  (1,2,3,...)**(4,9,16,...):  k*(k^3 + 8*k^2 + 23*k + 16)/12
%C A212891 row 3,  (1,2,3,...)**(9,16,25,...):  k*(k^3 + 12*k^2 + 53*k + 42)/12
%C A212891 ...
%C A212891 For a guide to related arrays, see A213500.
%F A212891 T(n,k) = 5*T(n,k-1) - 10*T(n,k-2) + 10*T(n,k-3) - 5*T(n,k-4) + T(n,k-5).
%F A212891 G.f. for row n:  f(x)/g(x), where f(x) = n^2 - (2*n^2 - 2*n - 1)*x + ((n-1)^2)*x^2 and g(x) = (1 - x)^5.
%e A212891 Northwest corner (the array is read by falling antidiagonals):
%e A212891 1....6....20....50....105....196...336
%e A212891 4....17...46....100...190....329...532
%e A212891 9....34...84....170...305....504...784
%e A212891 16...57...134...260...450....721...1092
%e A212891 25...86...196...370...625....980...1456
%e A212891 ...
%e A212891 T(5,1) = (1)**(25) = 25
%e A212891 T(5,2) = (1,2)**(25,36) = 1*36+2*25 = 86
%e A212891 T(5,3) = (1,2,3)**(25,36,49) = 1*49+2*36+3*25 = 196
%t A212891 b[n_] := n; c[n_] := n^2
%t A212891 t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
%t A212891 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A212891 Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
%t A212891 r[n_] := Table[t[n, k], {k, 1, 60}]  (* A212891 *)
%t A212891 d = Table[t[n, n], {n, 1, 40}] (* A213436 *)
%t A212891 s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
%t A212891 s1 = Table[s[n], {n, 1, 50}] (* A024166  *)
%Y A212891 Cf. A213500.
%K A212891 nonn,easy,tabl
%O A212891 1,2
%A A212891 _Clark Kimberling_, Jun 16 2012