A035029 - OEIS

%I #42 Oct 20 2022 03:34:56

%S 1,5,26,138,743,4043,22180,122468,679757,3789297,21199998,118973550,

%T 669447123,3775577367,21336790152,120795829128,684962855705,

%U 3889578815453,22115533878178,125892252068498,717400693313471,4092099111728355,23362391663233196,133488737662062188,763310051648602213

%N a(n) = Sum_{k=0..n} (k+1) * Sum_{j=0..n} 2^j*binomial(n,j)*binomial(n-k,j).

%C 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

%H Vincenzo Librandi, <a href="/A035029/b035029.txt">Table of n, a(n) for n = 0..1000</a>

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>

%H D. Merlini, R. Sprugnoli and M. C. Verri, <a href="http://dx.doi.org/10.1016/j.dam.2003.11.012">Waiting patterns for a printer</a>, FUN with algorithm'01, Isola d'Elba, 2001.

%F a(n) = (1/4)*(A002002(n+2) - A002002(n+1)).

%F G.f.: (1-z)^2/(8*z^2*sqrt(1-6*z+z^2))-(1+z)/(8*z^2). - _Emeric Deutsch_, Dec 28 2003

%F a(n) = T(n+1, n+2), array T as in A049600.

%F 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

%F 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). - _R. J. Mathar_, Dec 03 2012

%F a(n) ~ (3+2*sqrt(2))^(n+3/2) / (2^(9/4)*sqrt(Pi*n)). - _Vaclav Kotesovec_, Feb 12 2014

%F a(n) = Jypergeometric2F1([-n, n+3]; [1]; -1), which satisfies the recurrence. - _Benedict W. J. Irwin_, Oct 14 2016

%t 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 *)

%t With[{P = LegendreP}, Table[((n+1)*(n+3)*P[n+3,3] -(6*n^2+22*n+17)*P[n+2,3] +(n+ 2)*(5*n+8)*P[n+1,3])/(8*(n+1)*(n+2)), {n,0,40}]] (* _G. C. Greubel_, Oct 20 2022 *)

%o (Magma) I:=[1,5,26]; [n le 3 select I[n] else ( (7*n+1)*Self(n-1) - (7*n-15)*Self(n-2) + (n-3)*Self(n-3) )/(n+1): n in [1..30]]; // _G. C. Greubel_, Oct 20 2022

%o (SageMath)

%o def A001850(n): return gen_legendre_P(n,0,3)

%o def A035029(n): return ((n+1)*(n+3)*A001850(n+3) - (6*n^2 +22*n +17)*A001850(n+2) + (n+2)*(5*n+8)*A001850(n+1))/(8*(n+1)*(n+2))

%o [A035029(n) for n in range(40)] # _G. C. Greubel_, Oct 20 2022

%Y Cf. A001003, A001850, A002002, A049600.

%K nonn

%O 0,2

%A _N. J. A. Sloane_