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Rectangular array: (row n) = b**c, where b(h) = h, c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution.
4

%I #12 Jun 21 2012 12:17:47

%S 1,6,4,20,17,9,50,46,34,16,105,100,84,57,25,196,190,170,134,86,36,336,

%T 329,305,260,196,121,49,540,532,504,450,370,270,162,64,825,816,784,

%U 721,625,500,356,209,81,1210,1200,1164,1092,980,830,650,454,262

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

%C Principal diagonal: A213436

%C Antidiagonal sums: A024166

%C row 1, (1,2,3,...)**(1,4,9,...): A002415(k+1)

%C row 2, (1,2,3,...)**(4,9,16,...): k*(k^3 + 8*k^2 + 23*k + 16)/12

%C row 3, (1,2,3,...)**(9,16,25,...): k*(k^3 + 12*k^2 + 53*k + 42)/12

%C ...

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

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

%e 1....6....20....50....105....196...336

%e 4....17...46....100...190....329...532

%e 9....34...84....170...305....504...784

%e 16...57...134...260...450....721...1092

%e 25...86...196...370...625....980...1456

%e ...

%e T(5,1) = (1)**(25) = 25

%e T(5,2) = (1,2)**(25,36) = 1*36+2*25 = 86

%e T(5,3) = (1,2,3)**(25,36,49) = 1*49+2*36+3*25 = 196

%t b[n_] := n; c[n_] := n^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}] (* A212891 *)

%t d = Table[t[n, n], {n, 1, 40}] (* A213436 *)

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

%t s1 = Table[s[n], {n, 1, 50}] (* A024166 *)

%Y Cf. A213500.

%K nonn,easy,tabl

%O 1,2

%A _Clark Kimberling_, Jun 16 2012