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A064641 Unidirectional 'Delannoy' variation of the Boustrophedon transform applied to all 1's sequence: construct an array in which the first element of each row is 1 and subsequent entries are given by T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k) + T(n-2,k-1). The last number in row n gives a(n).

%I A064641 #94 Jul 18 2025 08:29:54
%S A064641 1,2,7,29,133,650,3319,17498,94525,520508,2910895,16487795,94393105,
%T A064641 545337200,3175320607,18615098837,109783526821,650884962908,
%U A064641 3877184797783,23193307022861,139271612505361,839192166392276,5072534905324615,30749397292689194
%N A064641 Unidirectional 'Delannoy' variation of the Boustrophedon transform applied to all 1's sequence: construct an array in which the first element of each row is 1 and subsequent entries are given by T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k) + T(n-2,k-1). The last number in row n gives a(n).
%C A064641 Also the number of paths from (0,0) to (n,n) not rising above y=x, using steps (1,0), (0,1), (1,1) and (2,1). For example, the 7 paths to (2,2) are dd, den, end, enen, Dn, eenn and edn, where e=(1,0), n=(0,1), d=(1,1) and D=(2,1). - _Brian Drake_, Aug 01 2007
%C A064641 For another interpretation as the number of walks of a certain type, see A223092 and the link below. - _N. J. A. Sloane_, Mar 29 2013
%C A064641 Hankel transform is 3^C(n+1,2). - _Paul Barry_, Jan 26 2009
%H A064641 Seiichi Manyama, <a href="/A064641/b064641.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Vincenzo Librandi)
%H A064641 Paul Barry, <a href="https://arxiv.org/abs/1802.03443">On a transformation of Riordan moment sequences</a>, arXiv:1802.03443 [math.CO], 2018.
%H A064641 Paul Barry, <a href="https://www.emis.de/journals/JIS/VOL22/Barry3/barry422.html">Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles</a>, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
%H A064641 Paul Barry, <a href="https://arxiv.org/abs/2001.08799">Characterizations of the Borel triangle and Borel polynomials</a>, arXiv:2001.08799 [math.CO], 2020.
%H A064641 P. Barry and A. Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry1/barry202.html">Four-term Recurrences, Orthogonal Polynomials and Riordan Arrays</a>, Journal of Integer Sequences, 2012, article 12.4.2. - From _N. J. A. Sloane_, Sep 21 2012
%H A064641 Brian Drake, <a href="http://dx.doi.org/10.1016/j.disc.2008.11.020">Limits of areas under lattice paths</a>, Discrete Math. 309 (2009), no. 12, 3936-3953.
%H A064641 M. Dziemianczuk, <a href="http://dx.doi.org/10.1007/s00373-013-1357-1">Counting Lattice Paths With Four Types of Steps</a>, Graphs and Combinatorics, September 2013, Volume 30, Issue 6, pp 1427-1452.
%H A064641 M. Dziemianczuk, <a href="http://arxiv.org/abs/1410.5747">On Directed Lattice Paths With Additional Vertical Steps</a>, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
%H A064641 N. J. A. Sloane, <a href="/A223092/a223092.pdf">Illustration of initial terms of A223092 and A064641</a>
%H A064641 D. V. Kruchinin, <a href="http://dx.doi.org/10.1186/s13662-014-0347-9">On solving some functional equations</a>, Advances in Difference Equations (2015) 2015:17; DOI 10.1186/s13662-014-0347-9.
%H A064641 <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>
%F A064641 G.f.: (1-x-sqrt(1-6x-3x^2)) / (2x(1+x)). - _Brian Drake_, Aug 01 2007
%F A064641 G.f.: 1/(1-2x-3x^2/(1-3x-3x^2/(1-3x-3x^2/(1-3x-3x^2/(1-.... (continued fraction). - _Paul Barry_, Jan 26 2009
%F A064641 a(n) = sum(i=0..n, binomial(n+i,n)*sum(j=0..n+1, binomial(j,-n+2*j-i-2)*binomial(n+1,j)))/(n+1). - _Vladimir Kruchinin_, May 12 2011
%F A064641 Recurrence: (n+1)*a(n) = (5*n-4)*a(n-1) + 9*(n-1)*a(n-2) + 3*(n-2)*a(n-3). - _Vaclav Kotesovec_, Oct 13 2012
%F A064641 a(n) ~ 3*(sqrt(6)+sqrt(2))*(3+2*sqrt(3))^n/(8*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 13 2012
%F A064641 G.f.: 1 / (1 - x - (x+x^2) / (1 - x - (x+x^2) / ... )) (continued fraction). - _Michael Somos_, Mar 30 2014
%F A064641 0 = a(n)*(+9*a(n+1) + 54*a(n+2) + 33*a(n+3) - 12*a(n+4)) + a(n+1)*(+78*a(n+2) + 60*a(n+3) - 27*a(n+4)) + a(n+2)*(+36*a(n+2) + 34*a(n+3) - 14*a(n+4)) + a(n+3)*(+4*a(n+3) + a(n+4)) for all n >= 0. - _Michael Somos_, Nov 05 2014
%F A064641 a(n) = (-1)^n * (n+1) + Sum_{k=0..n-1} (a(k) + (-1)^k) * (a(n-1-k) + (-1)^(n-1-k)). - _Seiichi Manyama_, Jul 18 2025
%e A064641 The array begins
%e A064641         1
%e A064641       1   2
%e A064641     1   5   7
%e A064641   1   8  22  29
%e A064641 G.f. = 1 + 2*x + 7*x^3 + 29*x^4 + 133*x^5 + 650*x^6 + 3319*x^7 + ...
%p A064641 A:= series( (1-x-sqrt(1-6*x-3*x^2)) / (2*x*(1+x)),x, 21): seq(coeff(A,x,i), i=0..20); # _Brian Drake_, Aug 01 2007
%t A064641 Table[SeriesCoefficient[(1-x-Sqrt[1-6*x-3*x^2])/(2*x*(1+x)),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 13 2012 *)
%o A064641 (PARI) a(n)=if(n<0,0,polcoeff(serreverse(x*(1-x)/(1+x+x^2)+O(x^(n+2))),n+1)) /* _Paul Barry_ */
%o A064641 (Maxima)
%o A064641 a(n):=sum(binomial(n+i,n)*sum(binomial(j,-n+2*j-i-2)*binomial(n+1,j),j,0,n+1),i,0,n)/(n+1); /* _Vladimir Kruchinin_, May 12 2011 */
%Y A064641 Delannoy numbers: A008288, table: A064642. Cf. A038764, A223092.
%Y A064641 Row sums of A201159.
%K A064641 nonn
%O A064641 0,2
%A A064641 _Floor van Lamoen_, Oct 03 2001