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 A039717 Row sums of convolution triangle A030523. 13
 1, 4, 15, 55, 200, 725, 2625, 9500, 34375, 124375, 450000, 1628125, 5890625, 21312500, 77109375, 278984375, 1009375000, 3651953125, 13212890625, 47804687500, 172958984375, 625771484375, 2264062500000, 8191455078125 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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) p-INVERT of (2^n), where p(s) = 1 - s - s^2; see A289780. - Clark Kimberling, Aug 10 2017 LINKS Michel Marcus, Table of n, a(n) for n = 1..1000 Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4. Index entries for linear recurrences with constant coefficients, signature (5,-5). FORMULA 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*(3*sqrt(5)/10+1/2)-(5/2-sqrt(5)/2)^n*(3*sqrt(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)(2*cos(r*Pi/10))^(2n)). a(n) = 5*a(n-1) - 5*a(n-2). a(n) = Sum_{k=0..n} Sum_{i=0..n} binomial(n, i)*binomial(k+i+1, 2k+1). - Paul Barry, Jun 22 2004 From Johannes W. Meijer, Jul 01 2010: (Start) Lim_{k->infinity} a(n+k)/a(k) = (A020876(n) + A093131(n)*sqrt(5))/2. Lim_{n->infinity} A020876(n)/A093131(n) = sqrt(5). (End) From Benito van der Zander, Nov 19 2015: (Start) Lim_{k->infinity} a(k+1)/a(k) = 1 + phi^2 = (5 + sqrt(5)) / 2. a(n) = a(n-1) * 3 + A081567(n-2) (not proved). (End) MATHEMATICA CoefficientList[Series[(1 - x) / (1 - 5 x + 5 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 10 2014 *) PROG (PARI) Vec(x*(1-x)/(1-5*x+5*x^2) + O(x^40)) \\ Altug Alkan, Nov 20 2015 CROSSREFS 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: A291029 A126932 A094833 * A220948 A026013 A050183 Adjacent sequences:  A039714 A039715 A039716 * A039718 A039719 A039720 KEYWORD easy,nonn AUTHOR STATUS approved

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Last modified October 15 22:25 EDT 2019. Contains 328038 sequences. (Running on oeis4.)