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A110122 Number of Delannoy paths of length n with no EE's crossing the line y = x (i.e., no two consecutive E steps from the line y = x+1 to the line y = x-1).

%I A110122 #45 Jan 05 2025 19:51:38
%S A110122 1,3,12,53,247,1192,5897,29723,152020,786733,4111295,21661168,
%T A110122 114925697,613442227,3291704108,17745496453,96062011319,521943400056,
%U A110122 2845404909129,15558847792747,85311186002036,468951179698653,2583765541267647,14266052382826208
%N A110122 Number of Delannoy paths of length n with no EE's crossing the line y = x (i.e., no two consecutive E steps from the line y = x+1 to the line y = x-1).
%C A110122 A Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps E=(1,0), N=(0,1) and D=(1,1).
%C A110122 Column 0 of A110121.
%H A110122 Nathaniel Johnston, <a href="/A110122/b110122.txt">Table of n, a(n) for n = 0..500</a>
%H A110122 P. Fahr and C. M. Ringel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Fahr/ringel44.html">A partition formula for Fibonacci Numbers</a>, JIS 11 (2008) Article 08.1.4.
%H A110122 Pedro Fernando Fernández Espinosa and Agustín Moreno Cañadas, <a href="https://arxiv.org/abs/2104.00050">Homological Ideals as Integer Specializations of Some Brauer Configuration Algebras</a>, arXiv:2104.00050 [math.RT], 2021.
%H A110122 M. D. Hirschhorn, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Hirschhorn/hirschhorn8.html">On Recurrences of Fahr and Ringel Arising in Graph Theory </a>, JIS 12 (2009) 09.6.8
%H A110122 Harris Kwong, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/48-4/Kwong.pdf">On recurrences of Fahr and Ringel: an alternate approach</a>, Fibonacci Quart. 48 (2010), no. 4, 363-365.
%H A110122 H. Prodinger, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Prodinger/prodinger24.html">Generating functions related to partition formulas for Fibonacci Numbers</a>, JIS 11 (2008) Article 08.1.8.
%H A110122 Robert A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Sulanke/delannoy.html">Objects Counted by the Central Delannoy Numbers</a>, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.
%F A110122 G.f.: 1/((1-zR)^2-z), where R = 1 + zR + zR^2 = (1 - z - sqrt(1 - 6z + z^2))/(2z) is the g.f. of the large Schroeder numbers (A006318).
%F A110122 a(n) = (1/(n+1))Sum_{k=0..n} (k+1) * Sum_{i=0..n-k} binomial(n+1, i)*binomial(2*n-k-i, n) * A000045(k+1). - _Vladimir Kruchinin_, Apr 18 2011
%F A110122 Recurrence: (2*n^2+9*n+7)*a(n) - (26*n^2+135*n+151)*a(n+1) + (88*n^2+528*n+746)*a(n+2) - (26*n^2+177*n+277)*a(n+3) + (2*n^2+15*n+25)*a(n+4)=0. - _Vaclav Kotesovec_, Sep 08 2012
%F A110122 a(n) ~ (10+7*sqrt(2))*sqrt((3*sqrt(2)-4)/Pi) * (3+2*sqrt(2))^n/n^(3/2). - _Vaclav Kotesovec_, Dec 11 2012
%e A110122 a(2) = 12 because, among the 13 (=A001850(2)) Delannoy paths of length 2, only NEEN has an EE crossing the line y=x.
%p A110122 R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=1/((1-z*R)^2-z): Gser:=series(G,z=0,27): 1,seq(coeff(Gser,z^n),n=1..24);
%t A110122 Flatten[{1, RecurrenceTable[{(2*n^2+9*n+7)*a[n]-(26*n^2+135*n+151) *a[n+1]+(88*n^2+528*n+746)*a[n+2]-(26*n^2+177*n+277)*a[n+3]+(2*n^2+15*n+25)*a[n+4]==0,a[1]==3,a[2]==12,a[3]==53,a[4]==247}, a, {n, 25}]}] (* _Vaclav Kotesovec_, Sep 09 2012 *)
%o A110122 (Maxima)
%o A110122 a(n):=sum((k+1)/(n+1)*sum(binomial(n+1,i)*binomial(2*n-k-i,n),i,0,n-k) *fib(k+1),k,0,n); /* _Vladimir Kruchinin_, Apr 18 2011 */
%Y A110122 Cf. A001850, A110121.
%K A110122 nonn
%O A110122 0,2
%A A110122 _Emeric Deutsch_, Jul 13 2005
%E A110122 Minor edits by _Vaclav Kotesovec_, Mar 31 2014