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 A035029 a(n) = Sum_{k=0..n} (k+1) * Sum_{j=0..n} 2^j*binomial(n,j)*binomial(n-k,j). 4
 1, 5, 26, 138, 743, 4043, 22180, 122468, 679757, 3789297, 21199998, 118973550, 669447123, 3775577367, 21336790152, 120795829128, 684962855705, 3889578815453, 22115533878178, 125892252068498, 717400693313471 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Number of regions in all the dissections of a convex (n+3)-gon by non-intersecting diagonals. a(1)=5 because in the three dissections of a square we have altogether five regions: one in the "no-diagonals" dissection and two in each of the dissections by one of the two diagonals of the square. - Emeric Deutsch, Dec 28 2003 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Milan Janjic, Two Enumerative Functions D. Merlini, R. Sprugnoli and M. C. Verri, Waiting patterns for a printer, FUN with algorithm'01, Isola d'Elba, 2001. FORMULA G.f.: (1-z)^2/(8*z^2*sqrt(1-6*z+z^2))-(1+z)/(8*z^2). - Emeric Deutsch, Dec 28 2003 a(n) = T(n+1, n+2), array T as in A049600. Form an array with the m(n,1)=1 and m(1,n) = n*(n+1)/2 for n=1,2,3... The interior terms m(i,j) = m(i,j-1) + m(i-1,j-1) + m(i-1,j). The terms in the main diagonal = {a(n)}. - J. M. Bergot, Dec 01 2012 D-finite with recurrence: (n+2)*a(n) + (-7*n-8)*a(n-1) + (7*n-8)*a(n-2) + (-n+2)*a(n-3) = 0. - R. J. Mathar, Dec 03 2012 a(n) ~ (3+2*sqrt(2))^(n+3/2) / (2^(9/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 12 2014 Conjecture: a(n) = 2F1(-n,n+3; 1; -1), which would satisfy above recurrence. - Benedict W. J. Irwin, Oct 14 2016 MATHEMATICA CoefficientList[Series[(1-x)^2/(8*x^2*Sqrt[1-6*x+x^2])-(1+x)/(8*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *) CROSSREFS Equals (1/4) [A002002(n+1) - A002002(n)]. Cf. A001003. Sequence in context: A288785 A161731 A049607 * A081569 A005573 A081911 Adjacent sequences:  A035026 A035027 A035028 * A035030 A035031 A035032 KEYWORD nonn AUTHOR STATUS approved

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Last modified November 29 13:17 EST 2021. Contains 349416 sequences. (Running on oeis4.)