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 A213751 Rectangular array:  (row n) = b**c, where b(h) = 2*h-1, c(h) = n-1+h, n>=1, h>=1, and ** = convolution. 5
 1, 5, 2, 14, 9, 3, 30, 23, 13, 4, 55, 46, 32, 17, 5, 91, 80, 62, 41, 21, 6, 140, 127, 105, 78, 50, 25, 7, 204, 189, 163, 130, 94, 59, 29, 8, 285, 268, 238, 199, 155, 110, 68, 33, 9, 385, 366, 332, 287, 235, 180, 126, 77, 37, 10, 506, 485, 447, 396, 336, 271 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Principal diagonal:  A051662 Antidiagonal sums:  A006325 row 1,  (1,3,5,7,9,...)**(1,2,3,4,5,...): A000330 row 2,  (1,3,5,7,9,...)**(2,3,4,5,6,...): A101986 row 3,  (1,3,5,7,9,...)**(3,4,5,6,7,...): (2*k^3 + 15*k^2 + k)/6 row 4,  (1,3,5,7,9,...)**(4,5,6,7,8,...): (2*k^3 + 21*k^2 + k)/6 For a guide to related arrays, see A213500. LINKS FORMULA T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4). G.f. for row n:  f(x)/g(x), where f(x) = n + x - (n - 1)*x^2 and g(x) = (1 - x )^4. EXAMPLE Northwest corner (the array is read by falling antidiagonals): 1...5....14...30....55....91 2...9....23...46....80....127 3...13...32...62....105...163 4...17...41...78....130...199 5...21...50...94....155...235 6...25...59...110...180...271 MATHEMATICA b[n_] := 2 n - 1; c[n_] := n; 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}]  (* A213751 *) Table[t[n, n], {n, 1, 40}] (* A051662 *) s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}] Table[s[n], {n, 1, 50}] (* A006325 *) CROSSREFS Cf. A213500. Sequence in context: A104634 A194008 A060422 * A185781 A265434 A189236 Adjacent sequences:  A213748 A213749 A213750 * A213752 A213753 A213754 KEYWORD nonn,tabl,easy AUTHOR Clark Kimberling, Jun 20 2012 STATUS approved

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Last modified April 21 19:16 EDT 2021. Contains 343156 sequences. (Running on oeis4.)