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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. 4
 1, 6, 4, 20, 17, 9, 50, 46, 34, 16, 105, 100, 84, 57, 25, 196, 190, 170, 134, 86, 36, 336, 329, 305, 260, 196, 121, 49, 540, 532, 504, 450, 370, 270, 162, 64, 825, 816, 784, 721, 625, 500, 356, 209, 81, 1210, 1200, 1164, 1092, 980, 830, 650, 454, 262 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Principal diagonal:  A213436 Antidiagonal sums:  A024166 row 1,  (1,2,3,...)**(1,4,9,...):  A002415(k+1) row 2,  (1,2,3,...)**(4,9,16,...):  k*(k^3 + 8*k^2 + 23*k + 16)/12 row 3,  (1,2,3,...)**(9,16,25,...):  k*(k^3 + 12*k^2 + 53*k + 42)/12 ... For a guide to related arrays, see A213500. LINKS FORMULA 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). 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. EXAMPLE Northwest corner (the array is read by falling antidiagonals): 1....6....20....50....105....196...336 4....17...46....100...190....329...532 9....34...84....170...305....504...784 16...57...134...260...450....721...1092 25...86...196...370...625....980...1456 ... T(5,1) = (1)**(25) = 25 T(5,2) = (1,2)**(25,36) = 1*36+2*25 = 86 T(5,3) = (1,2,3)**(25,36,49) = 1*49+2*36+3*25 = 196 MATHEMATICA b[n_] := n; c[n_] := n^2 t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}] TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]] Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]] r[n_] := Table[t[n, k], {k, 1, 60}]  (* A212891 *) d = Table[t[n, n], {n, 1, 40}] (* A213436 *) s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}] s1 = Table[s[n], {n, 1, 50}] (* A024166  *) CROSSREFS Cf. A213500. Sequence in context: A213761 A160248 A317858 * A107983 A009278 A213573 Adjacent sequences:  A212888 A212889 A212890 * A212892 A212893 A212894 KEYWORD nonn,easy,tabl AUTHOR Clark Kimberling, Jun 16 2012 STATUS approved

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Last modified May 11 21:35 EDT 2021. Contains 343808 sequences. (Running on oeis4.)