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A038731 Number of columns in all directed column-convex polyominoes of area n+1.

%I A038731 #67 Mar 06 2025 08:44:28
%S A038731 1,3,10,32,99,299,887,2595,7508,21526,61251,173173,486925,1362627,
%T A038731 3797374,10543724,29180067,80521055,221610563,608468451,1667040776,
%U A038731 4558234018,12441155715,33900136297,92230468249,250570010499,679844574322,1842280003640
%N A038731 Number of columns in all directed column-convex polyominoes of area n+1.
%C A038731 Apply Riordan array (1/(1-x), x/(1-x)^2) to n+1. - _Paul Barry_, Oct 13 2009
%C A038731 Binomial transform of (A001629 shifted left twice). - _R. J. Mathar_, Feb 06 2010
%H A038731 Vincenzo Librandi, <a href="/A038731/b038731.txt">Table of n, a(n) for n = 0..200</a>
%H A038731 Elena Barcucci, Renzo Pinzani, and Renzo Sprugnoli , <a href="http://dx.doi.org/10.1007/3-540-56610-4_71">Directed column-convex polyominoes by recurrence relations</a>, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.
%H A038731 Éva Czabarka, Rigoberto Flórez, Leandro Junes, and José L. Ramírez, <a href="https://doi.org/10.1016/j.disc.2018.06.032">Enumeration of peaks and valleys on non-decreasing Dyck paths</a>, Disc. Math. 341(10) (2018), 2789-2807; see Theorem 2 on p. 2791.
%H A038731 Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Florez/florez51.html">Counting Subwords in Non-Decreasing Dyck Paths</a>, Journal of Integer Sequences, Vol. 28 (2025), Article 25.1.6. See pp. 5, 15, 17, 19.
%H A038731 Jesus Salas and Alan D. Sokal, <a href="http://arxiv.org/abs/0711.1738">Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial</a>, J. Stat. Phys. 135 (2009) 279-373, arXiv:0711.1738. Mentions this sequence. - _N. J. A. Sloane_, Mar 14 2014
%H A038731 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6,-1).
%F A038731 5*a(n) = (2n+1)*F(2n+2) - (n-4)*F(2n+1), where the F(n)'s are the Fibonacci numbers, F(0)=0, F(1)=1.
%F A038731 a(n) = Sum_{k=1..n+1} k*binomial(n+k-1, 2k-2). - _Emeric Deutsch_, Jun 11 2003
%F A038731 From _Paul Barry_, Oct 13 2009: (Start)
%F A038731 G.f.: (1-x)^3/(1-3x+x^2)^2.
%F A038731 a(n) = Sum_{k=0..n} binomial(n+k, 2k)*(k+1). (End)
%F A038731 a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4). - _R. J. Mathar_, Feb 06 2010
%F A038731 a(n) = Sum_{k=0..n} (F(2k)+0^k)*F(2n-2k+1). - _Paul Barry_, Jun 23 2010
%F A038731 E.g.f.: exp(3*x/2)*(5*(5 + 4*x)*cosh(sqrt(5)*x/2) + sqrt(5)*(7 + 10*x)*sinh(sqrt(5)*x/2))/25. - _Stefano Spezia_, Mar 04 2025
%t A038731 Table[Sum[Binomial[n, k]*CoefficientList[Series[1/(1 - x - x^2)^2, {x, 0, k}], x][[-1]], {k, 0, n}], {n, 0, 27}] (* _Arkadiusz Wesolowski_, Feb 03 2012 *)
%t A038731 LinearRecurrence[{6, -11, 6, -1}, {1, 3, 10, 32}, 30] (* _Vincenzo Librandi_, Feb 04 2012 *)
%o A038731 (Magma) I:=[1, 3, 10, 32]; [n le 4 select I[n] else 6*Self(n-1)-11*Self(n-2)+6*Self(n-3)-Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Feb 04 2012
%o A038731 (Haskell)
%o A038731 a038731 n = a038731_list !! n
%o A038731 a038731_list = c [1] $ tail a000045_list where
%o A038731    c us vs'@(v:vs) = (sum $ zipWith (*) us vs') : c (v:us) vs
%o A038731 -- _Reinhard Zumkeller_, Oct 31 2013
%Y A038731 Row-sums of array T as in A038730.
%Y A038731 First differences of A030267.
%Y A038731 Row sums of A318942(n+1).
%Y A038731 Cf. A000045.
%K A038731 nonn,easy
%O A038731 0,2
%A A038731 _Clark Kimberling_, May 02 2000
%E A038731 Entry improved by comments from _Emeric Deutsch_, Jun 14 2001