%I A001850 M2942 N1184
%S A001850 1,3,13,63,321,1683,8989,48639,265729,1462563,8097453,45046719,
%T A001850 251595969,1409933619,7923848253,44642381823,252055236609,
%U A001850 1425834724419,8079317057869,45849429914943,260543813797441
%N A001850 Central Delannoy numbers: Sum_{k=0..n} C(n,k)*C(n+k,k).
%C A001850 Number of paths from (0,0) to (n,n) in an n X n grid using only steps
North, Northeast and East.
%C A001850 Also the number of ways of aligning two sequences (e.g. of nucleotides
or amino-acids) of length n, with at most 2*n gaps (-) inserted,
so that while unnecessary gappings: - -a a- - are forbidden, both
b- and -b are allowed. (If only other of the latter is allowed, then
the sequence A000984 gives the number of alignments). There is an
easy bijection from grid walks given by Dickau to such set of alignments
(e.g. the straight diagonal corresponds to the perfect alignment
with no gaps). - Antti Karttunen, Oct 10 2001
%C A001850 Also main diagonal of array defined by m(i,1)=m(1,j)=1, m(i,j)=m(i-1,
j-1)+m(i-1,j)+m(i,j-1) - Benoit Cloitre (benoit7848c(AT)orange.fr),
May 03 2002
%C A001850 a(n) is the number of n-matchings of a comb-like graph with 2n teeth.
Example: a(2)=13 because the graph consisting of a horizontal path
ABCD and the teeth Aa, Bb, Cc, Dd has 13 2-matchings: any of the
six possible pairs of teeth and {Aa, BC}, {Aa, CD}, {Bb, CD}, {Cc,
AB}, {Dd, AB}, {Dd, BC}, {AB, CD}. - Emeric Deutsch (deutsch(AT)duke.poly.edu),
Jul 02 2002
%C A001850 Number of ordered trees with 2n+1 edges, having root of odd degree, nonroot
nodes of outdegree at most 2 and branches of odd length. - Emeric
Deutsch (deutsch(AT)duke.poly.edu), Aug 02 2002
%C A001850 The sum of the first n coefficients of ((1-x)/(1-2x))^n is a(n-1). -
Michael Somos, Sep 28 2003
%C A001850 Row sums of A063007 and A105870. - Paul Barry (pbarry(AT)wit.ie), Apr
23 2005
%C A001850 Prime central Delannoy numbers include a(1) = 3, a(2) = 13, a(8) = 265729.
Note that these begin a(2^0), a(2^1), a(2^3). Semiprime central Delannoy
numbers include a(4) = 321 = 3 * 107, a(6) = 8989 = 89 * 101 - Jonathan
Vos Post (jvospost3(AT)gmail.com), May 22 2005
%C A001850 The Hankel transform (see A001906 for definition) of this sequence is
A036442 : 1, 4, 32, 512, 16384, ... . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Jul 03 2005
%C A001850 Also number of paths from (0,0) to (n,0) using only steps U=(1,1), H=(1,
0) and D=(1,-1), U can have 2 colors and H can have 3 colors. - Nour-Eddine
Fahssi (fahssin(AT)yahoo.fr), Jan 27 2008
%C A001850 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 30 2008:
(Start)
%C A001850 Equals row sums of triangle A152250 and INVERT transform of A109980:
%C A001850 (1, 2, 8, 36, 172, 852,...). (End)
%C A001850 Samieinia, p.3. The number of Khalimsky-continuous functions with codomain
Z can give an example of Delannoy numbers. If we consider instead
such functions with codomain N then we get an example of the other
numbers which are called the Schroeder numbers, [From Jonathan Vos
Post (jvospost3(AT)gmail.com), Mar 27 2009]
%D A001850 J.-M. Autebert et al., Le treillis des Chemins de Delannoy, Discrete
Math., 258 (2002), 225-234.
%D A001850 J.-M. Autebert et al., On generalized Delannoy paths, SIAM J. Discrete
Math., 16 (2003), 208-223.
%D A001850 Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal
Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%D A001850 Frits Beukers, Arithmetic properties of Picard-Fuchs equations, S\'eminaire
de Th\'eorie des nombres de Paris, 1982-83, Birkh"auser Boston, Inc.
%D A001850 J. S. Caughman et al., A note on lattice chains and Delannoy numbers,
Discrete Math., 308 (2008), 2623-2628.
%D A001850 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
%D A001850 M. D. Hirschhorn, How many ways can a king cross the board?, Austral.
Math. Soc. Gaz., 27 (2000), 104-106.
%D A001850 S. Kaparthi and H. R. Rao, Higher dimensional restricted lattice paths
with diagonal steps, Discr. Appl. Math., 31 (1991), 279-289.
%D A001850 D. F. Lawden, On the Solution of Linear Difference Equations, Math. Gaz.,
36 (1952), 193-196.
%D A001850 L. Moser, King paths on a chessboard, Note 2487, Math. Gaz., 39 (1955)
54 (one page only).
%D A001850 L. Moser and W. Zayachkowski, Lattice paths with diagonal steps, Scripta
Math., 26 (1961), 223-229.
%D A001850 Munarini, Emanuele, Combinatorial properties of the antichains of a garland.
Integers, 9 (2009), 353-374.
%D A001850 Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients,
Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
%D A001850 Shiva Samieinia, The number of continuous curves in digital geometry,
Stockholm University, Research Reports in Mathematics, Number 3,
December 22, 2008. [From Jonathan Vos Post (jvospost3(AT)gmail.com),
Mar 27 2009]
%D A001850 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A001850 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A001850 R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 2, 1999; see
Example 6.3.8 and Problem 6.49.
%D A001850 R. G. Stanton and D. D. Cowan, Note on a "square" functional equation,
SIAM Rev., 12 (1970), 277-279.
%D A001850 R. A. Sulanke et al., Another description of the central Delannoy numbers,
Problem 10894, Amer. Math. Monthly, 110 (No. 5, 2003), 443-444.
%H A001850 T. D. Noe, <a href="b001850.txt">Table of n, a(n) for n = 0..200</a>
%H A001850 C. Banderier and S. Schwer, <a href="http://arXiv.org/abs/math.CO/0411128">
Why Delannoy numbers?</a>
%H A001850 E. Deutsch and B. E. Sagan, <a href="http://arxiv.org/pdf/math.CO/0407326">
Congruences for Catalan and Motzkin numbers and related sequences</
a>, J. Num. Theory 117 (2006), 191-215.
%H A001850 R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/delannoy.html">
Delannoy and Motzkin Numbers</a> [Many illustrations]
%H A001850 R. M. Dickau, <a href="a001850.gif">The 13 paths in a 4 X 4 grid</a>
%H A001850 Nick Hobson, <a href="a001850.py.txt">Python program for this sequence</
a>
%H A001850 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative
Functions</a>
%H A001850 P. Peart and W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/
JIS/index.html">Generating Functions via Hankel and Stieltjes Matrices</
a>, J. Integer Seqs., Vol. 3 (2000), #00.2.1.
%H A001850 R. Pemantle and M. C. Wilson, <a href="http://arXiv.org/abs/math.CO/0003192">
Asymptotics of multivariate sequences, I: smooth points of the singular
variety</a>
%H A001850 Shiva Samieinia, <a href="http://www.math.su.se/reports/2008/3">The number
of continuous curves in digital geometry</a>, December 22, 2008.
[From Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 27 2009]
%H A001850 T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/
series006">The Delannoy numbers</a>
%H A001850 R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Moments of generalized Motzkin paths</a>, J. Integer Sequences, Vol.
3 (2000), #00.1.
%H A001850 R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Objects Counted by the Central Delannoy Numbers </a>, Journal of
Integer Sequences, Vol. 5 (2002), Article 03.1.5
%H A001850 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
DelannoyNumber.html">Link to a section of The World of Mathematics.</
a>
%H A001850 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
SchmidtsProblem.html">Schmidt's Problem</a>
%H A001850 W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Hankel Matrices and Lattice Paths</a>, J. Integer Sequences, 4 (2001),
#01.1.2.
%F A001850 P_n(3), where P_n is n-th Legendre polynomial.
%F A001850 G.f.: 1/sqrt(1-6*x+x^2).
%F A001850 a(n) =a(n-1)+2*A002002(n) =sum{j}[A063007(n, j)] - Henry Bottomley (se16(AT)btinternet.com),
Jul 02 2001
%F A001850 Dominant term in asymptotic expansion is binomial(2n, n)/2^(1/4)*((sqrt(2)+1)/
2)^(2n+1)*(1+c_1/n+c_2/n^2+ ... ) - Mike Hirschhorn (mikeh(AT)maths.unsw.edu.au)
%F A001850 a(n) = Sum_{i=0..n} (A000079[i]*A008459[n, i]) = Sum_{i=0..n} (2^i *
C(n, i)^2). - Antti Karttunen, Oct 10 2001
%F A001850 a(n) = sum(k=0, n, C(n+k, n-k)*C(2k, k) ). - Benoit Cloitre (benoit7848c(AT)orange.fr),
Feb 13 2003
%F A001850 a(n)=sum(k=0, n, C(n, k)^2*2^k). - Michael Somos, Oct 08 2003
%F A001850 E.g.f.: exp(3*x)*BesselI(0, 2*sqrt(2)*x). - Vladeta Jovovic (vladeta(AT)eunet.rs),
Mar 21 2004
%F A001850 a(n)=sum{k=0..n, C(2n-k, n)C(n, k)}; - Paul Barry (pbarry(AT)wit.ie),
Apr 23 2005
%F A001850 a(n) = Sum_{k>=n} binomial(k, n)^2/2^(k+1). - Vladeta Jovovic (vladeta(AT)eunet.rs),
Aug 25 2006
%F A001850 a(-1-n)=a(n). - Michael Somos Sep 23 2006
%F A001850 a(-1) = a(0) = 1; n*a(n) = 3*(2*n-1)*a(n-1) - (n-1)*a(n-2).
%F A001850 Comment from Peter John (peter.john(AT)tu-ilmenau.de), Oct 19 2006: Define
general Delannoy numbers by (i,j>0): d(i,0)=d(0,j)=1=:d(0,0)and d(i,
j)=d(i-1,j-1)+d(i-2,j-1)+d(i-1,j). Then a(k)= sum[{d(k,j)^2}+{d(k-1,
j)^2}], j>=0. This is a special case of the following formula for
general Delannoy numbers: d(k,j)=sum[{d(p,i)*d(n-p,j-i)}+{d(p-1,i)*d(n-p-1,
j-i-1)}], where i>=0 and p=0,1,...,n.
%F A001850 Coefficient of x^n in (1 + 3*x + 2*x^2)^n. - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr),
Jan 11 2008
%F A001850 a(n)=A008288(A046092(n)). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Apr 08 2009]
%F A001850 G.f.: 1/(1-x-2x/(1-x-x/(1-x-x/(1-x-x/(1-... (continued fraction). [From
Paul Barry (pbarry(AT)wit.ie), May 28 2009]
%p A001850 seq(add(multinomial(n+k,n-k,k,k),k=0..n),n=0..20); - Zerinvary Lajos
(zerinvarylajos(AT)yahoo.com), Oct 18 2006
%t A001850 {Dela[ 0 ], Dela[ 1 ]} = {1, 3}; Dela[ n_ ] := Dela[ n ] = (3(2n-1) Dela[
n-1 ] - (n-1) Dela[ n-2 ])/n
%o A001850 (PARI) {a(n)=if(n<0, n=-1-n); polcoeff(1/sqrt(1-6*x+x^2+x*O(x^n)), n)}
/* Michael Somos Sep 23 2006 */
%o A001850 (PARI) {a(n)=if(n<0, n=-1-n); subst(pollegendre(n), x, 3)} /* Michael
Somos Sep 23 2006 */
%o A001850 (PARI) {a(n)=if(n<0, n=-1-n); n++; subst(Pol(((1-x)/(1-2*x)+O(x^n))^n),
x, 1)} /* Michael Somos Sep 23 2006 */
%o A001850 (Python program from Nick Hobson. Replace leading dots by spaces)
%o A001850 .def f(a, b):
%o A001850 .... if a == 0 or b == 0: return 1
%o A001850 .... return f(a, b-1) + f(a-1, b) + f(a-1, b-1)
%o A001850 ....
%o A001850 .for n in xrange(7):
%o A001850 .... print f(n, n),
%o A001850 ....
%o A001850 (PARI) a(n)=if(n<0, 0, polcoeff((1+3x+2x^2)^n, n)) - Paul Barry (pbarry(AT)wit.ie),
Aug 22 2007
%Y A001850 Cf. A008288, A027618, A047665. a(n)=T(n, n-1), array T as in A049600.
%Y A001850 Main diagonal of A064861.
%Y A001850 Cf. A026003, A052141, A084773.
%Y A001850 A152250, A109980 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 30
2008]
%Y A001850 Cf. A000129, A078057 [From Jonathan Vos Post (jvospost3(AT)gmail.com),
Mar 27 2009]
%Y A001850 Sequence in context: A092467 A034478 A026715 this_sequence A130525 A000259
A007855
%Y A001850 Adjacent sequences: A001847 A001848 A001849 this_sequence A001851 A001852
A001853
%K A001850 nonn,easy,nice
%O A001850 0,2
%A A001850 N. J. A. Sloane (njas(AT)research.att.com).
%E A001850 New name and reference Sep 15 1995. Formula and more references from
D. E. Knuth May 15 1996.
%E A001850 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 16 2000
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