A005555 -id:A005555 - cp's OEIS frontend

A005573 Number of walks on cubic lattice (starting from origin and not going below xy plane).

1, 5, 26, 139, 758, 4194, 23460, 132339, 751526, 4290838, 24607628, 141648830, 817952188, 4736107172, 27487711752, 159864676803, 931448227590, 5435879858958, 31769632683132, 185918669183370, 1089302293140564

Offset: 0

N. J. A. Sloane

Binomial transform of A026378, second binomial transform of A001700. - Philippe Deléham, Jan 28 2007

The Hankel transform of [1,1,5,26,139,758,...] is [1,4,15,56,209,...](see A001353). - Philippe Deléham, Apr 13 2007

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Emeric Deutsch and Jim Brawner, Problem 10795: Three-Dimensional Lattice Walks in the Upper Half-Space, Amer. Math. Monthly, 108 (Dec. 2001), 980.
Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Further Results on Paths in an n-Dimensional Cubic Lattice, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.2.
R. K. Guy, Letter to N. J. A. Sloane, May 1990
R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 97.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (Sqrt((1-2*x)/(1-6*x)) -1)/(2*x) )); // G. C. Greubel, May 02 2019

CoefficientList[Series[(Sqrt[(1-2x)/(1-6x)]-1)/(2x),{x,0,20}],x] (* Harvey P. Dale, Jun 24 2011 *)
a[n_] := 6^n Hypergeometric2F1[1/2, -n, 2, 2/3]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 11 2017 *)

my(x='x+O('x^30)); Vec((sqrt((1-2*x)/(1-6*x)) -1)/(2*x)) \\ G. C. Greubel, May 02 2019

((sqrt((1-2*x)/(1-6*x)) -1)/(2*x)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 02 2019

From Emeric Deutsch, Jan 09 2003; corrected by Roland Bacher: (Start)
a(n) = Sum_{i=0..n} (-1)^i*6^(n-i)*binomial(n, i)*binomial(2*i, i)/(i+1);
g.f. A(x) satisfies: x(1-6x)A^2 + (1-6x)A - 1 = 0. (End)
From Henry Bottomley, Aug 23 2001: (Start)
a(n) = 6*a(n-1) - A005572(n-1).
a(n) = Sum_{j=0..n} 4^(n-j)*binomial(n, floor(n/2))*binomial(n, j). (End)
a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2*k+1, k)*2^(n-k).
a(n) = Sum_{k=0..n} (-1)^k*binomial(n, k)*Catalan(k)*6^(n-k).
D-finite with recurrence (n+1)*a(n) = (8*n+2)*a(n-1)-(12*n-12)*a(n-2). - Vladeta Jovovic, Jul 16 2004
a(n) = Sum_{k=0..n} A052179(n,k). - Philippe Deléham, Jan 28 2007
Conjecture: a(n)= 6^n * hypergeom([1/2,-n],[2], 2/3). - Benjamin Phillabaum, Feb 20 2011
From Paul Barry, Apr 21 2009: (Start)
G.f.: (sqrt((1-2*x)/(1-6*x)) - 1)/(2*x).
G.f.: 1/(1-5*x-x^2/(1-4*x-x^2/(1-4*x-x^2/(1-4*x-x^2/(1-... (continued fraction). (End)
G.f.: 1/(1 - 4*x - x*(1 - 2*x)/(1 - 2*x - x*(1 - 2*x)/(1 - 2*x - x*(1 - 2*x)/(1 - 2*x - x*(1 - 2*x)/(1...(continued fraction). - Aoife Hennessy (aoife.hennessy(AT)gmail.com), Jul 02 2010
a(n) ~ 6^(n+1/2)/sqrt(Pi*n). - Vaclav Kotesovec, Oct 05 2012
G.f.: G(0)/(2*x) - 1/(2*x), where G(k)= 1 + 4*x*(4*k+1)/( (4*k+2)*(1-2*x) - 2*x*(1-2*x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1-2*x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 24 2013
a(n) = 2^n*hypergeom([-n, 3/2], [2], -2). - Peter Luschny, Apr 26 2016
E.g.f.: exp(4*x)*(BesselI(0,2*x) + BesselI(1,2*x)). - Ilya Gutkovskiy, Sep 20 2017

More terms from Henry Bottomley, Aug 23 2001

A005564 Number of n-step walks on square lattice in the first quadrant which finish at distance n-3 from the x-axis.

Original entry on oeis.org

6, 20, 45, 84, 140, 216, 315, 440, 594, 780, 1001, 1260, 1560, 1904, 2295, 2736, 3230, 3780, 4389, 5060, 5796, 6600, 7475, 8424, 9450, 10556, 11745, 13020, 14384, 15840, 17391, 19040, 20790, 22644, 24605, 26676, 28860, 31160, 33579, 36120, 38786, 41580, 44505

Offset: 3

N. J. A. Sloane

The steps are N, S, E or W.
For n>=4, a(n-1)/2 is the coefficient c(n-2) of the m^(n-2) term of P(m,n) = (c(m-1)* m^(n-1) + c(m-2)* m^(n-2) +...+ c(0)* m^0)/((a!)* (a-1)!), the polynomial for the number of partitions of m with exactly n parts. - Gregory L. Simay, Jun 28 2016
2a(n) is the denominator of formula 207 in Jolleys' "Summation of Series." 2/(1*3*4)+3/(2*4*5)+...n terms. Sum_{k = 1..n} (k+1)/(k*(k+2)*(k+3)). This summation has a closed form of 17/36-(6*n^2+21*n+17)/(6*(n+1)*(n+2)*(n+3)). - Gary Detlefs, Mar 15 2018
a(n) is the number of degrees of freedom in a tetrahedral cell for a Nédélec first kind finite element space of order n-2. - Matthew Scroggs, Jan 02 2021

			The n=4 diagram in Fig. 4 of Guy's paper is:
1
0 4
9 0 6
0 16 0 4
10 0 9 0 1
Adding 16+4 we get a(4)=20.
The a(3) = 6 walks are EEN, ENE, ENW, NEW, NSN, NNS. - _Michael Somos_, Jun 09 2014
G.f. = 6*x^3 + 20*x^4 + 45*x^5 + 84*x^6 + 140*x^7 + 216*x^8 + 315*x^9 + ...
From _Gregory L. Simay_ Jun 28 2016: (Start)
P(m,4) = (m^3 + 3*m^2 + ...)/(3!*4!) with 3 = a(3)/2 = 6/2.
P(m,5) = (m^4 + 10*m^3 + ...)/(4!*5!) with 10 = a(4)/2 = 20/2.
P(m,6) = (m^5 + (45/2)*m^4 +...)/(5!*6!) with 45/2 = a(5)/2.
P(m,7) = (m^6 + 42*m^5 +...)/(6!*7!) with 42 = a(6)/2 = 84/2. (End)

L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 38.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
DefElement, Nédélec first kind
R. K. Guy, Letter to N. J. A. Sloane, May 1990
R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6. See figure 4, sum of terms in (n-2)-nd row.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000217.
First differences of A001701.
Fourth column of A093768.

a:=List([0..45],n->(n+1)*Binomial(n+4,2)); # Muniru A Asiru, Feb 15 2018

I:=[6, 20, 45, 84]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..45]]; // Vincenzo Librandi, Jun 18 2012

A005564 := proc(n)
        (n-2)*(n)*(n+1)/2 ;
end proc: seq(A005564(n),n=0..10) ; # R. J. Mathar, Dec 09 2011

Table[(n-2)*Binomial[n+1, 2], {n, 3, 40}]
LinearRecurrence[{4,-6,4,-1},{6,20,45,84},50] (* Vincenzo Librandi, Jun 18 2012 *)

a(n)=(n-2)*(n)*(n+1)/2 \\ Charles R Greathouse IV, Dec 12 2011

G.f.: x^3 * ( 6 - 4*x + x^2 ) / ( 1 - x )^4. [Simon Plouffe in his 1992 dissertation]
a(n) = (n-2)*n*(n+1)/2 = (n-2)*A000217(n).
a(n) = Sum_{j = 0..n} ((n+j-1)^2-(n-j+1)^2)/4. - Zerinvary Lajos, Sep 13 2006
a(n) = Sum_{k = 2..n-1} k*n. - Zerinvary Lajos, Jan 29 2008
a(n) = 4*binomial(n+1,2)*binomial(n+1,4)/binomial(n+1,3) = (n-2)*binomial(n+1,2). - Gary Detlefs, Dec 08 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 18 2012
E.g.f.: x - x*(2 - 2*x - x^2)*exp(x)/2. - Ilya Gutkovskiy, Jun 29 2016
a(n) = 6*Sum_{i = 1..n-1} A000217(i) - n*A000217(n). - Bruno Berselli, Jul 03 2018
Sum_{n>=3} 1/a(n) = 5/18. - Amiram Eldar, Oct 07 2020

Entry revised by N. J. A. Sloane, Jul 06 2012

A005556 Exponents m_i associated with Weyl group W(E6).

Original entry on oeis.org

1, 4, 5, 7, 8, 11

Offset: 1

N. J. A. Sloane

H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 141.

R. K. Guy, Letter to N. J. A. Sloane, May 1990

Exponents(RootDatum("E6")); // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

Edited by N. J. A. Sloane, May 13 2008

A005557 Number of walks on square lattice.

Original entry on oeis.org

42, 132, 297, 572, 1001, 1638, 2548, 3808, 5508, 7752, 10659, 14364, 19019, 24794, 31878, 40480, 50830, 63180, 77805, 95004, 115101, 138446, 165416, 196416, 231880, 272272, 318087, 369852, 428127, 493506, 566618, 648128, 738738, 839188, 950257, 1072764

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Richard K. Guy, Letter to N. J. A. Sloane, May 1990.
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Sixth diagonal of Catalan triangle A033184.
Sixth column of Catalan triangle A009766.
Cf. A062991, A214292.

List([0..30],n->(n+1)*Binomial(n+10,4)/5); # Muniru A Asiru, Apr 10 2018

[(n+1)*Binomial(n+10, 4)/5: n in [0..40]]; // Vincenzo Librandi, Mar 20 2013

[seq(binomial(n,5)-binomial(n,3),n=9..55)]; # Zerinvary Lajos, Jul 19 2006
A005557:=(42-120*z+135*z**2-70*z**3+14*z**4)#(z-1)**6; # conjectured by Simon Plouffe in his 1992 dissertation

CoefficientList[Series[(14 z^4 - 70 z^3 + 135 z^2 - 120 z + 42)/(z - 1)^6, {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 22 2011 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{42,132,297,572,1001,1638},40] (* Harvey P. Dale, Feb 22 2024 *)

a(n)=(n+1)*binomial(n+10,4)/5 \\ Charles R Greathouse IV, Oct 21 2022

a(n) = A009766(n+5, 5) = (n+1)*binomial(n+10, 4)/5.
G.f.: (42 - 120*x + 135*x^2 - 70*x^3 + 14*x^4)/(1-x)^6; numerator polynomial is N(2;4, x) from A062991.
a(n) = binomial(n+9,5) - binomial(n+9,3). - Zerinvary Lajos, Jul 19 2006
a(n) = A214292(n+9, 4). - Reinhard Zumkeller, Jul 12 2012
From Amiram Eldar, Sep 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 2509/63504.
Sum_{n>=0} (-1)^n/a(n) = 951395/63504 - 1360*log(2)/63. (End)

More terms and formula from Wolfdieter Lang, Sep 04 2001

A005560 Number of walks on square lattice. Column y=2 of A052174.

Original entry on oeis.org

1, 3, 15, 45, 189, 588, 2352, 7560, 29700, 98010, 382239, 1288287, 5010005, 17177160, 66745536, 232092432, 901995588, 3173688180, 12342120700, 43861998180, 170724392916, 611947174608, 2384209771200, 8609646396000, 33577620944400, 122041737663300, 476432168185575

Offset: 2

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
R. K. Guy, Letter to N. J. A. Sloane, May 1990
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6, w_n'(2).

Cf. A005558, A005559, A005561, A005562, A093768.

Cf. A052174.

[Binomial(n+3, Ceiling(n/2))*Binomial(n+2, Floor(n/2)) - Binomial(n+3, Ceiling((n-1)/2))*Binomial(n+2, Floor((n-1)/2)): n in [0..30]]; // Vincenzo Librandi, Apr 03 2017

wnprime := proc(n,y)
    local k;
    if type(n-y,'even') then
        k := (n-y)/2 ;
        binomial(n+1,k)*(binomial(n,k)-binomial(n,k-1)) ;
    else
        k := (n-y-1)/2 ;
        binomial(n+1,k)*binomial(n,k+1)-binomial(n+1,k+1)*binomial(n,k-1) ;
    end if;
end proc:
A005560 := proc(n)
    wnprime(n,2) ;
end proc:
seq(A005560(n),n=2..20) ; # R. J. Mathar, Apr 02 2017

Table[Binomial[n+3, Ceiling[n/2]] Binomial[n+2, Floor[n/2]]-Binomial[n+3, Ceiling[(n-1)/2]] Binomial[n+2, Floor[(n-1)/2]], {n, 0, 30}] (* Vincenzo Librandi, Apr 03 2017 *)

{a(n)=binomial(n+3,ceil(n/2))*binomial(n+2,floor(n/2)) - binomial(n+3,ceil((n-1)/2))*binomial(n+2,floor((n-1)/2))}

a(n) = C(n+3, ceiling(n/2))*C(n+2, floor(n/2)) - C(n+3, ceiling((n-1)/2))*C(n+2, floor((n-1)/2)). - Paul D. Hanna, Apr 16 2004

Conjecture: (n-1)*(n-2)*(2*n+1)*(n+5)*(n+4)*a(n) -4*n*(n+1)*(2*n^2+4*n+19)*a(n-1) -16*n^2*(n-1)*(2*n+3)*(n+1)*a(n-2)=0. - R. J. Mathar, Apr 02 2017

A005559 Number of walks on square lattice. Column y=1 of A052174.

Original entry on oeis.org

1, 2, 8, 20, 75, 210, 784, 2352, 8820, 27720, 104544, 339768, 1288287, 4294290, 16359200, 55621280, 212751396, 734959368, 2821056160, 9873696560, 38013731756, 134510127752, 519227905728, 1854385377600, 7174705330000, 25828939188000, 100136810390400

Offset: 1

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. K. Guy, Letter to N. J. A. Sloane, May 1990
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6 (see figure 5).

Cf. A005558-A005560, A093768.

Cf. A052174.

[Binomial(n+2, Ceiling(n/2))*Binomial(n+1, Floor(n/2)) - Binomial(n+2, Ceiling((n-1)/2))*Binomial(n+1, Floor((n-1)/2)): n in [0..30]]; // Vincenzo Librandi, Oct 16 2014

seq(binomial(n+1, ceil((n-1)/2))*binomial(n, floor((n-1)/2)) -binomial(n+1, ceil((n-2)/2))*binomial(n, floor((n-2)/2)), n=1..30); # Robert Israel, Oct 19 2014

Table[Binomial[n+2, Ceiling[n/2]] Binomial[n+1, Floor[n/2]] - Binomial[n+2, Ceiling[(n-1)/2]] Binomial[n+1, Floor[(n-1)/2]], {n, 0, 200}] (* Vincenzo Librandi, Oct 17 2014 *)

{a(n)=binomial(n+2,ceil(n/2))*binomial(n+1,floor(n/2)) - binomial(n+2,ceil((n-1)/2))*binomial(n+1,floor((n-1)/2))}

a(n) = C(n+1,ceiling((n-1)/2)) *C(n,floor((n-1)/2)) -C(n+1,ceiling((n-2)/2)) *C(n,floor((n-2)/2)). - Paul D. Hanna, Apr 16 2004
G.f.: -(48*x^3-16*x^2-3*x+1)*EllipticK(4*x)/(12*Pi*x^4)+(4*x^2-9*x+1)*EllipticE(4*x)/(12*Pi*x^4)+1/(4*x^3)-1/(2*x^2) (using Maple's convention for elliptic integrals: EllipticE(t) = Integral_{s=0..1} sqrt(1 - s^2*t^2)/sqrt(1-s^2) ds, EllipticK(t) = Integral_{s=0..1} ((1-s^2*t^2)*(1-s^2))^(-1/2) ds). - Robert Israel, Oct 19 2014
Conjecture: -(n-1)*(2*n+1)*(n+4)*(n+3)*a(n) +4*(n+1)*(2*n^2+4*n+9)*a(n-1) +16*n*(n-1)*(2*n+3)*(n+1)*a(n-2)=0. - R. J. Mathar, Apr 02 2017

A005561 Number of walks on square lattice. Column y=3 of A052174.

Original entry on oeis.org

1, 4, 24, 84, 392, 1344, 5760, 19800, 81675, 283140, 1145144, 4008004, 16032016, 56632576, 225059328, 801773856, 3173688180, 11392726800, 44986664800, 162594659920, 641087516256, 2331227331840, 9183622822400, 33577620944400, 132211882468575, 485773975404900

Offset: 3

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
R. K. Guy, Letter to N. J. A. Sloane, May 1990
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6. See w_n'(3).

Cf. A005558, A005559, A005560, A005562, A093768.

Cf. A052174.

[Binomial(n+4, Ceiling(n/2))*Binomial(n+3, Floor(n/2)) - Binomial(n+4, Ceiling((n-1)/2))*Binomial(n+3, Floor((n-1)/2)): n in [0..30]]; // Vincenzo Librandi, Apr 03 2017

wnprime := proc(n,y)
    local k;
    if type(n-y,'even') then
        k := (n-y)/2 ;
        binomial(n+1,k)*(binomial(n,k)-binomial(n,k-1)) ;
    else
        k := (n-y-1)/2 ;
        binomial(n+1,k)*binomial(n,k+1)-binomial(n+1,k+1)*binomial(n,k-1) ;
    end if;
end proc:
A005561 := proc(n)
    wnprime(n,3) ;
end proc:
seq(A005561(n),n=3..30) ; # R. J. Mathar, Apr 02 2017

Table[Binomial[n+4, Ceiling[n/2]] Binomial[n+3, Floor[n/2]]-Binomial[n+4, Ceiling[(n-1)/2]] Binomial[n+3, Floor[(n-1)/2]], {n, 0, 30}] (* Vincenzo Librandi, Apr 03 2017 *)

{a(n)=binomial(n+4,ceil(n/2))*binomial(n+3,floor(n/2)) - binomial(n+4,ceil((n-1)/2))*binomial(n+3,floor((n-1)/2))}

a(n) = C(n+4, ceiling(n/2))*C(n+3, floor(n/2)) - C(n+4, ceiling((n-1)/2))*C(n+3, floor((n-1)/2)). - Paul D. Hanna, Apr 16 2004

Conjecture: (n-2)*(n-3)*(2*n+1)*(n+6)*(n+5)*a(n) - 4*n*(n+1)*(2*n^2+4*n+33)*a(n-1) - 16*n^2*(n-1)*(2*n+3)*(n+1)*a(n-2) = 0. - R. J. Mathar, Apr 02 2017

A005562 Number of walks on square lattice. Column y=4 of A052174.

Original entry on oeis.org

1, 5, 35, 140, 720, 2700, 12375, 45375, 196625, 715715, 3006003, 10930920, 45048640, 164105760, 668144880, 2441298600, 9859090500, 36149998500, 145173803500, 534239596880, 2136958387520, 7892175863000, 31479019635375, 116657543354625, 464342770607625, 1726402608669375

Offset: 4

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 4..1000
R. K. Guy, Letter to N. J. A. Sloane, May 1990
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6, w_n'(4).

Cf. A005558, A005559, A005560, A005561, A093768.

Cf. A052174.

[Binomial(n+5, Ceiling(n/2))*Binomial(n+4, Floor(n/2)) - Binomial(n+5, Ceiling((n-1)/2))*Binomial(n+4, Floor((n-1)/2)): n in [0..30]]; // Vincenzo Librandi, Apr 03 2017

wnprime := proc(n,y)
    local k;
    if type(n-y,'even') then
        k := (n-y)/2 ;
        binomial(n+1,k)*(binomial(n,k)-binomial(n,k-1)) ;
    else
        k := (n-y-1)/2 ;
        binomial(n+1,k)*binomial(n,k+1)-binomial(n+1,k+1)*binomial(n,k-1) ;
    end if;
end proc:
A005562 := proc(n)
    wnprime(n,4) ;
end proc:
seq(A005562(n),n=4..30) ; # R. J. Mathar, Apr 02 2017

Table[Binomial[n+5, Ceiling[n/2]] Binomial[n+4, Floor[n/2]]-Binomial[n+5, Ceiling[(n-1)/2]] Binomial[n+4, Floor[(n-1)/2]], {n, 0, 30}] (* Vincenzo Librandi, Apr 03 2017 *)

{a(n)=binomial(n+5,ceil(n/2))*binomial(n+4,floor(n/2)) - binomial(n+5,ceil((n-1)/2))*binomial(n+4,floor((n-1)/2))}

a(n) = C(n+5, ceiling(n/2))*C(n+4, floor(n/2)) - C(n+5, ceiling((n-1)/2))*C(n+4, floor((n-1)/2)). - Paul D. Hanna, Apr 16 2004

Conjecture: (n-3)*(n-4)*(2*n+1)*(n+7)*(n+6)*a(n) - 4*n*(n+1)*(2*n^2+4*n+51)*a(n-1) - 16*n^2*(n-1)*(2*n+3)*(n+1)*a(n-2) = 0. - R. J. Mathar, Apr 02 2017

A005565 Number of walks on square lattice.

Original entry on oeis.org

20, 75, 189, 392, 720, 1215, 1925, 2904, 4212, 5915, 8085, 10800, 14144, 18207, 23085, 28880, 35700, 43659, 52877, 63480, 75600, 89375, 104949, 122472, 142100, 163995, 188325, 215264, 244992, 277695, 313565, 352800, 395604, 442187, 492765, 547560, 606800

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. K. Guy, Letter to N. J. A. Sloane, May 1990
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

[1/4*(n^4+14*n^3+69*n^2+136*n+80): n in [0..40]]; // Vincenzo Librandi, May 24 2012

seq(add (k^3-n^2, k =0..n), n=4..28 ); # Zerinvary Lajos, Aug 26 2007
A005565:=(-20+25*z-14*z**2+3*z**3)/(z-1)**5; # conjectured by Simon Plouffe in his 1992 dissertation

CoefficientList[Series[(20-25x+14x^2-3x^3)/(1-x)^5,{x,0,40}],x] (* Vincenzo Librandi, May 24 2012 *)
LinearRecurrence[{5,-10,10,-5,1},{20,75,189,392,720},40] (* Harvey P. Dale, Dec 04 2020 *)
Differences[Table[Sum[x^3 - y^2, {x, 0, g}, {y, x, g}], {g, 3, 30}]] (* Horst H. Manninger, Jun 19 2025 *)

a(n)=(n^4+14*n^3+69*n^2+136*n)/4+20 \\ Charles R Greathouse IV, Nov 22 2011

From Ralf Stephan, Apr 23 2004: (Start)
a(n) = (1/4)*(n^4+14n^3+69n^2+136n+80).
G.f.: (20-25x+14x^2-3x^3)/(1-x)^5. (End)
a(n) = binomial(n+4,2)^2 - binomial(n+4,1)^2. - Gary Detlefs, Nov 22 2011
Using two consecutive triangular numbers t(n) and t(n+1), starting at n=3, compute the determinant of a 2 X 2 matrix with the first row t(n), t(n+1) and the second row t(n+1), 2*t(n+1). This gives (n+1)^2*(n-2)*(n+2)/4 = a(n-3). - J. M. Bergot, May 17 2012
E.g.f.: exp(x)*(80 + 220*x + 118*x^2 + 20*x^3 + x^4)/4. - Stefano Spezia, Jun 20 2025

A090749 a(n) = 12 * C(2n+1,n-5) / (n+7).

Original entry on oeis.org

1, 12, 90, 544, 2907, 14364, 67298, 303600, 1332045, 5722860, 24192090, 100975680, 417225900, 1709984304, 6962078952, 28192122176, 113649492522, 456442180920, 1827459250276, 7297426411968, 29075683360185, 115631433392020, 459124809056550, 1820529677650320, 7210477496434485

Offset: 5

Philippe Deléham, Feb 15 2004

Also a diagonal of A059365 and A009766. See also A000108, A002057, A003517, A003518, A003519.

Number of standard tableaux of shape (n+6,n-5). - Emeric Deutsch, May 30 2004

G. C. Greubel, Table of n, a(n) for n = 5..1000
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, Bounce statistics for rational lattice paths, arXiv:1707.09918 [math.CO], 2017, p. 9.
Richard K. Guy, Letter to N. J. A. Sloane, May 1990.
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.

Cf. A001622, A009766, A039598, A059365, A214292.

Cf. A000108, A002057, A003517, A003518, A003519.

Table[12*Binomial[2*n + 1, n - 5]/(n + 7), {n,5,50}] (* G. C. Greubel, Feb 07 2017 *)

for(n=5,50, print1(12*binomial(2*n+1,n-5)/(n+7), ", ")) \\ G. C. Greubel, Feb 07 2017

a(n) = A039598(n, 5) = A033184(n+7, 12).
G.f.: x^5*C(x)^12 with C(x) g.f. of A000108(Catalan).
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=11, a(n-6)=(-1)^(n-11)*coeff(charpoly(A,x),x^11). - Milan Janjic, Jul 08 2010
a(n) = A214292(2*n,n-6) for n > 5. - Reinhard Zumkeller, Jul 12 2012
From Karol A. Penson, Nov 21 2016: (Start)
O.g.f.: z^5 * 4^6/(1+sqrt(1-4*z))^12.
Recurrence: (-4*(n-5)^2-58*n+80)*a(n+1)-(-n^2-6*n+27)*a(n+2)=0, a(0),a(1),a(2),a(3),a(4)=0,a(5)=1,a(6)=12, n>=5.
Asymptotics: (-903+24*n)*4^n*sqrt(1/n)/(sqrt(Pi)*n^2).
Integral representation as n-th moment of a signed function W(x) on x=(0,4),in Maple notation: a(n+5)=int(x^n*W(x),x=0..4),n=0,1,..., where W(x)=(256/231)*sqrt(4-x)*JacobiP(5, 1/2, 1/2, (1/2)*x-1)*x^(11/2)/Pi and JacobiP are Jacobi polynomials. Note that W(0)=W(4)=0. (End).
From Ilya Gutkovskiy, Nov 21 2016: (Start)
E.g.f.: 6*exp(2*x)*BesselI(6,2*x)/x.
a(n) ~ 3*2^(2*n+3)/(sqrt(Pi)*n^(3/2)). (End)
From Amiram Eldar, Sep 26 2022: (Start)
Sum_{n>=5} 1/a(n) = 88699/15120 - 71*Pi/(27*sqrt(3)).
Sum_{n>=5} (-1)^(n+1)/a(n) = 102638*log(phi)/(75*sqrt(5)) - 22194839/75600, where phi is the golden ratio (A001622). (End)

Missing term 113649492522 inserted by Ilya Gutkovskiy, Dec 07 2016

cp's OEIS Frontend

A005573 Number of walks on cubic lattice (starting from origin and not going below xy plane).

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A005564 Number of n-step walks on square lattice in the first quadrant which finish at distance n-3 from the x-axis.

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A005556 Exponents m_i associated with Weyl group W(E6).

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A005557 Number of walks on square lattice.

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A005560 Number of walks on square lattice. Column y=2 of A052174.

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A005559 Number of walks on square lattice. Column y=1 of A052174.

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