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A318944 Number of Dyck paths with n nodes and altitude 4.

%I A318944 #14 May 23 2024 11:56:35
%S A318944 0,0,0,0,1,8,32,114,376,1177,3549,10406,29861,84249,234502,645625,
%T A318944 1761765,4772534,12851261,34434561,91890118,244385617,648139821,
%U A318944 1714976054,4529163125,11942440233,31448759302,82727323369,217426319541,571033273142
%N A318944 Number of Dyck paths with n nodes and altitude 4.
%H A318944 Colin Barker, <a href="/A318944/b318944.txt">Table of n, a(n) for n = 0..1000</a>
%H A318944 E. Czabarka et al, <a href="https://doi.org/10.1016/j.disc.2018.06.032">Enumerations of peaks and valleys on non-decreasing Dyck paths</a>, Disc. Math. 341 (2018) 2789-2807.
%H A318944 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (9,-31,50,-36,8).
%F A318944 a(n) = 9*a(n-1) - 31*a(n-2) + 50*a(n-3) - 36*a(n-4) + 8*a(n-5) for n>9. - _Colin Barker_, Apr 11 2019
%p A318944 (1-x)^2*x^4*(1+x-8*x^2+7*x^3)/(1-2*x)^3/(1-3*x+x^2) ;
%p A318944 taylor(%,x=0,30) ;
%p A318944 gfun[seriestolist](%) ; # _R. J. Mathar_, Nov 25 2018
%t A318944 LinearRecurrence[{9,-31,50,-36,8},{0,0,0,0,1,8,32,114,376,1177},30] (* _Harvey P. Dale_, Nov 03 2019 *)
%o A318944 (PARI) concat([0,0,0,0], Vec(x^4*(1 - x)^2*(1 + x - 8*x^2 + 7*x^3) / ((1 - 2*x)^3*(1 - 3*x + x^2)) + O(x^40))) \\ _Colin Barker_, Apr 11 2019
%Y A318944 A column of A318942.
%K A318944 nonn,easy
%O A318944 0,6
%A A318944 _N. J. A. Sloane_, Sep 18 2018