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 A204201 Triangle based on (0,1/3,1) averaging array. 7
 1, 1, 4, 1, 5, 10, 1, 6, 15, 22, 1, 7, 21, 37, 46, 1, 8, 28, 58, 83, 94, 1, 9, 36, 86, 141, 177, 190, 1, 10, 45, 122, 227, 318, 367, 382, 1, 11, 55, 167, 349, 545, 685, 749, 766, 1, 12, 66, 222, 516, 894, 1230, 1434, 1515, 1534, 1, 13, 78, 288, 738, 1410 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS For a1, let t(n,1)=[a+t(n-1,1)]/2, t(n,n)=[b+t(n-1,n-1)]/2, t(n,k)=[t(n-1,k-1)+t(n-1,k)]/2 for 2<=k<=n-1. We call (t(n,k)) the (a,r,b) averaging array.  If a and b are integers and r is a rational number, then multiplying row n of (t(n,k)) by the LCM of its denominators yields a triangle of integers; A204201 arises in this manner from (a,r,b)=(0,1/3,1). ... Guide to related arrays: (a,r,b).........triangle (0,1/2,1).......A054143 (0,1/3,1).......A204201 (0,2/3,1).......A204202 (0,1/4,1).......A204203 (0,3/4,1).......A204204 (0,1/5,1).......A204205 (1,3/2,2).......A204206 (1,2,3).........A204207 LINKS FORMULA From Philippe Deléham, Dec 24 2013: (Start) T(n,n) = A033484(n-1). Sum{k=1..n} T(n,k) = A053220(n). T(n,k) = T(n-1,k)+3*T(n-1,k-1)-2*T(n-2,k-1)-2*T(n-2,k-2), T(1,1)=1, T(2,1)=1, T(2,2)=4, T(n,k)=0 if k<1 or if k>n. (End) EXAMPLE The (0,1/3,1) averaging array has these first four rows: 1/3 1/6....2/3 1/12...5/12...5/6 1/24...1/4....5/8...11/12. Multiplying those rows by 3,6,12,24, respectively: 1 1...4 1...5...10 1...6...15...22 The first nine rows: 1 1...4 1...5...10 1...6...15...22 1...7...21...37...46 1...8...28...58...83...94 1...9...36...86...141..177..190 1...10..45...122..227..318..367..382 1...11..55...167..349..545..685..749..766 MATHEMATICA a = 0; r = 1/3; b = 1; t[1, 1] = r; t[n_, 1] := (a + t[n - 1, 1])/2; t[n_, n_] := (b + t[n - 1, n - 1])/2; t[n_, k_] := (t[n - 1, k - 1] + t[n - 1, k])/2; u[n_] := Table[t[n, k], {k, 1, n}] Table[u[n], {n, 1, 5}]   (* averaging array *) u = Table[(1/2) (1/r) 2^n*u[n], {n, 1, 12}]; TableForm[u]             (* A204102 triangle *) Flatten[u]               (* A204201 sequence *) CROSSREFS Cf. A204202. Sequence in context: A153426 A261720 A080852 * A090842 A120868 A100279 Adjacent sequences:  A204198 A204199 A204200 * A204202 A204203 A204204 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Jan 12 2012 STATUS approved

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Last modified September 22 05:47 EDT 2019. Contains 327287 sequences. (Running on oeis4.)