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 A213783 Rectangular array:  (row n) = b**c, where b(h) = 1+[h/2], c(h) = [(n+h)/2], n>=1, h>=1, [ ] = floor, and ** = convolution. 4
 1, 3, 1, 6, 4, 2, 11, 8, 6, 2, 17, 14, 11, 7, 3, 26, 22, 19, 13, 9, 3, 36, 32, 28, 22, 16, 10, 4, 50, 45, 41, 33, 27, 18, 12, 4, 65, 60, 55, 47, 39, 30, 21, 13, 5, 85, 79, 74, 64, 56, 44, 35, 23, 15, 5, 106, 100, 94, 84, 74, 62, 50, 38, 26, 16, 6, 133, 126, 120, 108 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Principal diagonal: A213759. Antidiagonal sums: A213760. Row 1, (1,2,2,3,3,4,4,...)**(1,1,2,2,3,3,4,...): A005744. Row 2, (1,2,2,3,3,4,4,5,...)**(1,2,2,3,3,4,4,5,...). Row 3, (1,2,2,3,3,4,4,5,...)**(2,2,3,3,4,4,5,5,...). For a guide to related arrays, see A213500. LINKS Clark Kimberling, Antidiagonals n = 1..80, flattened FORMULA T(n,k) = 2*T(n,k-1) -T(n,k-2) - 4*T(n,k-3) +T(n,k-4) +2*T(n,k-5) -T(n,k-6). G.f. for row n:  f(x)/g(x), where f(x) = [(n+1)/2] + [(n+2)/2]*x + ([(n-1)/2] + [(n+1)/2])*x^2 - (1+[n/2]-(n mod 2))*x^3 + [n/2]*x^4 and g(x) = (1 + x)^2 *(1 - x)^4, where [ ] = floor. EXAMPLE Northwest corner (the array is read by falling antidiagonals): 1...3....6....11...17...26...36....50 1...4....8....14...22...32...45....60 2...6....11...19...28...41...55....74 2...7....13...22...33...47...64....84 3...9....16...27...39...56...74....98 3...10...18...30...44...62...83....108 4...12...21...35...50...71...93....122 4...13...23...38...55...77...102...132 MATHEMATICA b[n_] := Floor[(n + 2)/2]; c[n_] := Floor[(n + 1)/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}]  (* A213783 *) Table[t[n, n], {n, 1, 40}] (* A213759 *) s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}] Table[s[n], {n, 1, 50}] (* A213760 *) CROSSREFS Cf. A213500. Sequence in context: A121437 A078585 A116551 * A163330 A021320 A193864 Adjacent sequences:  A213780 A213781 A213782 * A213784 A213785 A213786 KEYWORD nonn,tabl,easy AUTHOR Clark Kimberling, Jun 22 2012 STATUS approved

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Last modified December 12 15:11 EST 2019. Contains 329960 sequences. (Running on oeis4.)