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 A213550 Rectangular array: (row n) = b**c, where b(h) = h*(h+1)/2, c(h) = n-1+h, n>=1, h>=1, and ** = convolution. 7
 1, 5, 2, 15, 9, 3, 35, 25, 13, 4, 70, 55, 35, 17, 5, 126, 105, 75, 45, 21, 6, 210, 182, 140, 95, 55, 25, 7, 330, 294, 238, 175, 115, 65, 29, 8, 495, 450, 378, 294, 210, 135, 75, 33, 9, 715, 660, 570, 462, 350, 245, 155, 85, 37, 10, 1001, 935, 825, 690, 546 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Principal diagonal: A002418 Antidiagonal sums: A005585 row 1, (1,3,6,...)**(1,2,3,...): A000332 row 2, (1,3,6,...)**(2,3,4,...): A005582 row 3, (1,3,6,...)**(3,4,5,...): A095661 row 4, (1,3,6,...)**(4,5,6,...): A095667 For a guide to related arrays, see A213500. LINKS Table of n, a(n) for n=1..60. 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-(n-1)*x and g(x) = (1 - x)^5. EXAMPLE Northwest corner (the array is read by falling antidiagonals): 1....5....15...35....70....126 2....9....25...55....105...182 3....13...35...75....140...238 4....17...45...95....175...294 5....21...55...115...210...350 MATHEMATICA b[n_] := n (n + 1)/2; 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}] (* A213550 *) d = Table[t[n, n], {n, 1, 40}] (* A002418 *) s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}] s1 = Table[s[n], {n, 1, 50}] (* A005585 *) CROSSREFS Cf. A213500, A213548. Sequence in context: A191722 A191435 A128142 * A283242 A246209 A297979 Adjacent sequences: A213547 A213548 A213549 * A213551 A213552 A213553 KEYWORD nonn,tabl,easy AUTHOR Clark Kimberling, Jun 16 2012 STATUS approved

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Last modified May 25 03:08 EDT 2024. Contains 372782 sequences. (Running on oeis4.)