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A039717
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Row sums of convolution triangle A030523.
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11
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1, 4, 15, 55, 200, 725, 2625, 9500, 34375, 124375, 450000, 1628125, 5890625, 21312500, 77109375, 278984375, 1009375000, 3651953125, 13212890625, 47804687500, 172958984375, 625771484375, 2264062500000, 8191455078125
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OFFSET
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1,2
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COMMENTS
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Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,....,2n, s(0) = 3, s(2n) = 5.
With offset 0 = INVERT transform of A001792: (1, 3, 8, 20, 48, 112,...). - Gary W. Adamson, Oct 26 2010
From Tom Copeland, Nov 09 2014: (Start)
The array belongs to a family of arrays associated to the Catalan A000108 (t=1), and Riordan, or Motzkin sums A005043 (t=0), with the o.g.f. [1-sqrt(1-4x/(1+(1-t)x))]/2 and inverse x(1-x)/[1+(t-1)x(1-x)]. See A091867 for more info on this family. Here t=-4 (mod signs in the results).
Let C(x) = [1 - sqrt(1-4x)]/2, an o.g.f. for the Catalan numbers A000108, with inverse Cinv(x) = x*(1-x) and P(x,t) = x/(1+t*x) with inverse P(x,-t).
O.g.f.: G(x) = x*(1-x)/[1 - 5x*(1-x)] = P[Cinv(x),-5].
Inverse O.g.f.: Ginv(x) = [1 - sqrt(1 - 4*x/(1+5x))]/2 = C[P(x,5)] (signed A026378). Cf. A030528. (End)
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LINKS
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Table of n, a(n) for n=1..24.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
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FORMULA
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G.f.: x*(1-x)/(1-5*x+5*x^2)= g1(3, x)/(1-g1(3, x)), g1(3, x) := x*(1-x)/(1-2*x)^2 (G.f. first column of A030523).
Binomial transform of Fib(2n+2). a(n)=(sqrt(5)/2+5/2)^n(3sqrt(5)/10+1/2)-(5/2-sqrt(5)/2)^n(3sqrt(5)/10-1/2). - Paul Barry, Apr 16 2004
a(n) = (1/5)*Sum(r, 1, 9, Sin(3*r*Pi/10)Sin(r*Pi/2)(2Cos(r*Pi/10))^(2n)).
a(n) = 5a(n-1) - 5a(n-2).
a(n) = sum{k=0..n, sum{i=0..n, C(n, i)C(k+i+1, 2k+1)}}. - Paul Barry, Jun 22 2004
From Johannes W. Meijer, Jul 01 2010: (Start)
Limit(a(n+k)/a(k), k=infinity) = (A020876(n) + A093131(n)*sqrt(5))/2.
Limit(A020876(n)/A093131(n), n=infinity) = sqrt(5).
(End)
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MATHEMATICA
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CoefficientList[Series[(1 - x) / (1 - 5 x + 5 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 10 2014 *)
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CROSSREFS
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Cf. A000045.
Appears in A109106. - Johannes W. Meijer, Jul 01 2010
Cf. A001792. - Gary W. Adamson, Oct 26 2010
Cf. A000108, A005043, A091867, A026378, A030528.
Sequence in context: A219603 A126932 A094833 * A220948 A026013 A050183
Adjacent sequences: A039714 A039715 A039716 * A039718 A039719 A039720
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KEYWORD
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easy,nonn
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AUTHOR
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Wolfdieter Lang
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STATUS
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approved
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