A060150 -id:A060150 - cp's OEIS frontend

A254686 Number of ways to put n red and n blue balls into n indistinguishable boxes.

1, 1, 5, 19, 74, 248, 814, 2457, 7168, 19928, 53688, 139820, 354987, 878434, 2128102, 5052010, 11781881, 27019758, 61035671, 135928105, 298784144, 648726349, 1392474574, 2956730910, 6214668074, 12937060340, 26686392239, 54572423946, 110680119454, 222710856175, 444776676764

Offset: 0

Brian Chen, Feb 08 2015

nonn

See a comment on A254811 about multiset partitions and the Knuth reference. - Wolfdieter Lang, Mar 26 2015

			For n = 2 the a(2) = 5 ways to put 2 red balls and 2 blue balls into 2 indistinguishable boxes are (RRBB)(), (RRB)(B), (RBB)(R), (RR)(BB), (RB)(RB).

Brian Chen, Table of n, a(n) for n = 0..64

Cf. A002774, A060150, A254811.
Column k=2 of A256384.
Main diagonal of A277239.

with(numtheory):
b:= proc(n, k, i) option remember;
      `if`(n>k, 0, 1) +`if`(isprime(n) or i<2, 0, add(
      `if`(d>k, 0, b(n/d, d, i-1)), d=divisors(n) minus {1, n}))
    end:
a:= n-> b(6^n$2,n):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 26 2015

b[n_, k_, i_] := b[n, k, i] = If[n > k, 0, 1] + If[PrimeQ[n] || i < 2, 0, Sum[If[d > k, 0, b[n/d, d, i - 1]], {d, Divisors[n] [[2 ;; -2]]}]]; a[n_] := b[6^n, 6^n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)

A337902 The number of walks of length 2n+1 on the square lattice that start from the origin (0,0) and end at the vertex (2,1).

Original entry on oeis.org

3, 50, 735, 10584, 152460, 2208492, 32207175, 472780880, 6982113996, 103673813880, 1546866469148, 23179817220000, 348690679038000, 5263441096145400, 79698007774092375, 1210159553338375200, 18422202264818467500, 281089726445607849000

Offset: 1

R. J. Mathar, Sep 29 2020

			a(1)=3 represents 3 walks of length 3: RRU, URR and RUR.

Cf. A002894 (at (0,0)), A060150 (at (1,0)), A135389 (at (1,1)), A337900 (at (2,0)), A337901 (at (3,0))

a(n) =  binomial(2*n+1,n-1)*binomial(2*n+1,n) = A002054(n)*A001700(n).
G.f.: 3*x*3F2(2,5/2,5/2; 3,4; 16*x).
D-finite with recurrence  (n-1)*(n+2)*(n+1)*a(n) -4*n*(2*n+1)^2*a(n-1)=0.
A135389(n) = 2*A060150(n+1) +2*a(n).

A337869 The number of random walks on the simple square lattice that return to the origin (0,0) after 2n steps and do not pass through (0,0) or (1,0) at intermediate steps.

Original entry on oeis.org

3, 13, 106, 1073, 12142, 147090, 1865772, 24463905, 328887346, 4508608610, 62781858592, 885513974674, 12624162072740, 181611275997040, 2633023723495116, 38431604042148681, 564258290166041298, 8327627696761062714, 123471550301117915892

Offset: 1

R. J. Mathar, Sep 27 2020

The number of walks on the simple square lattice that take one of the four directions U, D, R, L at each step and return to zero is zero if the number of steps is odd. If the number of steps is even, the sequence counts walks that start at (0,0), return to (0,0) and never pass through (0,0) or (1,0) in between.

The ordinary generating function is a mix of inverses of sums and differences of the hypergeometric generating functions in A002894 and A060150. See Maple.

			Example: a(1)=3 counts the walks UD, DU, LR (but not RL which would pass (1,0)) of 2 steps that return to the origin.

R. J. Mathar, Random Walk on the Square Lattice: Return to (0,0) with or without passing (1,0) (Sep 2020)

Cf. A002894, A060150, A337870.

g002894 := hypergeom([1/2,1/2],[1],16*x^2) ;
g060150 := x*hypergeom([1,3/2,3/2],[2,2],16*x^2) ;
1-1/2/(g002894+g060150)-1/2/(g002894-g060150) ;
taylor(%,x=0,40);
gfun[seriestolist](%) ; # includes zeros of odd steps

A337901 The number of walks of length 2n+1 on the square lattice that start from the origin (0,0) and end at the vertex (3,0).

Original entry on oeis.org

1, 25, 441, 7056, 108900, 1656369, 25050025, 378224704, 5712638724, 86394844900, 1308887012356, 19868414760000, 302198588499600, 4605510959127225, 70321771565375625, 1075697380745222400, 16483023079048102500, 252980753801047064100, 3888662839165553120100

Offset: 1

R. J. Mathar, Sep 29 2020

Cf. A002894 (end at (0,0)), A060150 (end at (1,0)), A135389 (end at (1,1)), A337900 (at (2,0)), A337902 (at(2,1))

a(n) = [A002054(n)]^2.
G.f.: x*4F3(2,2,5/2,5/2; 1,4,4; 16*x).
D-finite with recurrence (n+2)^2*(n-1)^2*a(n) -4*n^2*(2*n+1)^2*a(n-1)=0.

A380241 Array read by antidiagonals: T(n,k) is the number of rooted (2k)-regular planar maps with n vertices, n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 9, 1, 1, 1, 14, 100, 54, 1, 1, 1, 42, 1225, 3000, 378, 1, 1, 1, 132, 15876, 171500, 110000, 2916, 1, 1, 1, 429, 213444, 10001880, 30012500, 4550000, 24057, 1, 1, 1, 1430, 2944656, 591666768, 7981500240, 5987493750, 204000000, 208494, 1, 1

Offset: 0

Andrew Howroyd, Jan 22 2025

The zeroth column is included by convention only for consistency with the first row sequences.

The case for regular planar maps of odd valency is more complicated and without simple closed form formulas, so not presented in this sequence. See the references for additional information.

			Array begins:
====================================================================
n\k | 0  1      2          3               4                   5 ...
----+---------------------------------------------------------------
  0 | 1  1      1          1               1                   1 ...
  1 | 1  1      2          5              14                  42 ...
  2 | 1  1      9        100            1225               15876 ...
  3 | 1  1     54       3000          171500            10001880 ...
  4 | 1  1    378     110000        30012500          7981500240 ...
  5 | 1  1   2916    4550000      5987493750       7304332956480 ...
  6 | 1  1  24057  204000000   1302227368750    7310748066293952 ...
  7 | 1  1 208494 9690000000 301107909375000 7794097754539041792 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
E. A. Bender and E. R. Canfield, The number of degree restricted rooted maps on the sphere, SIAM J. Discrete Math. 7 (1994) 9-15.
Zhicheng Gao and Mizan Rahman, Enumeration of k-poles, Annals of Combinatorics 1 (1997), pp. 55-66.
W. T. Tutte, A Census of Slicings, Canad. J. Math. 14 (1962), 708-722.
W. T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.

Columns 0..3 are A000012 twice, A000168, A380242.
Rows 0..3 are A000012, A000108, A060150, A380243.
Cf. A269920.

T(n,k)=if(k==0, 1, 2*binomial(2*k-1,k)^n*(n*k)!/(n!*(n*k - n + 2)!))

T(n,k) = 2*binomial(2*k-1, k)^n*(n*k)!/(n!*(n*k - n + 2)!) for k > 0.

A378060 a(n) = binomial(n, floor((n-1)/2))^2.

Original entry on oeis.org

0, 1, 1, 9, 16, 100, 225, 1225, 3136, 15876, 44100, 213444, 627264, 2944656, 9018009, 41409225, 130873600, 590976100, 1914762564, 8533694884, 28210561600, 124408576656, 418151049316, 1828114918084, 6230734868736, 27043120090000, 93271169290000, 402335398890000

Offset: 0

Peter Luschny, Dec 03 2024

Number of walks of length n with unit steps in all four directions (NSWE), starting at the origin and ending on the y-axis, never going below the x-axis and the end point having a positive height.

			The 16 walks of length 4: NNNN, NNNS, NNSN, NNEW, NNWE, NSNN, NENW, NEWN, NWNE, NWEN, ENNW, ENWN, EWNN, WNNE, WNEN, WENN.

Cf. A060150 (odd bisection), A337900 (even bisection), A037952, A378061.

# Generates the walks (for illustration only).
function aCount(n::Int)
    a = [""]
    c = 0
    for w in a
        if length(w) == n
            if (count('N', w) != count('S', w) && count('W', w) == count('E', w))
                c += 1
                # println(w)
            end
        else
            for j in "NSEW"
                u = string(w, j)
                if count('N', u) >= count('S', u)
                   push!(a, u)
    end end end end
    return c
end
println([aCount(n) for n in 0:11])

a := n -> binomial(n, iquo(n+1, 2) - 1)^2: seq(a(n), n = 0..27);
a := proc(n) option remember; if n < 2 then n else ((32*n^5 - 48*n^4 + 16*n^2)*a(n - 2) + (8*n^4 - 20*n^2)*a(n - 1))/(2*(n - 1)^2*(n - 1/2)*(n + 2)^2) fi end:
# Alternative:
egf := BesselI(0, 2*x)^2 + BesselI(1, 2*x)*BesselI(0, 2*x)*(1 - 1/x):
ser := series(egf, x, 29): seq(n!*coeff(ser, x, n), n = 0..27);

Array[Binomial[#, Floor[(# + 1)/2] - 1]^2 &, 28, 0] (* Michael De Vlieger, Dec 04 2024 *)

a(n) = n!*[x^n] (BesselI(0, 2*x)^2 + BesselI(1, 2*x)*BesselI(0, 2*x)*(1 - 1/x)).
a(n) = [x^n] (((8*x^2 + 2*x)*EllipticK(4*x) - Pi*(1 + x) + 2*EllipticE(4*x))/(4*x^2*Pi)).
a(n) = [x^n] (x*hypergeom([1,3/2,3/2], [2,2], 16*x^2) + x^2*hypergeom([3/2,3/2,2,2], [1,3,3], 16*x^2)).
a(n) = Sum_{k=0..n} (-1)^(n-k+N)*C(n-k, N)*C(n, k)*C(n+k, k), where N = floor((n-1)/2) and C = binomial.
Recurrence: a(n) = ((32*n^5 - 48*n^4 + 16*n^2)*a(n - 2) + (8*n^4 - 20*n^2)*a(n - 1))/(2*(n - 1)^2*(n - 1/2)*(n + 2)^2).
a(n) = Sum_{k=1..n} A378061(n, k).

A378070 a(n) = binomial(n - 1, ceiling(n/2)) * binomial(n - 1, ceiling(n/2) - 1).

Original entry on oeis.org

1, 0, 1, 2, 9, 24, 100, 300, 1225, 3920, 15876, 52920, 213444, 731808, 2944656, 10306296, 41409225, 147232800, 590976100, 2127513960, 8533694884, 31031617760, 124408576656, 456164781072, 1828114918084, 6749962774464, 27043120090000, 100445874620000, 402335398890000

Offset: 0

Peter Luschny, Dec 13 2024

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A007318, A060150 (even bisection), A135389 (odd bisection), A378060.

a :=  n -> binomial(n-1, floor((n+1)/2))*binomial(n-1, floor((n+1)/2)-1);
seq(a(n), n = 0..27);

A378070[n_] := Binomial[n - 1, #]*Binomial[n - 1, # - 1] & [Ceiling[n/2]];
Array[A378070, 30, 0] (* Paolo Xausa, Dec 14 2024 *)

a(n) = binomial(n - 1, floor(n/2) - 1) * binomial(n - 1, ceiling(n/2) - 1).

A116421 a(n) = 2^(n-1)*binomial(2n-1,n-1)^2.

Original entry on oeis.org

0, 1, 18, 400, 9800, 254016, 6830208, 188457984, 5300380800, 151289881600, 4369251780608, 127394382495744, 3743979352236032, 110768619888640000, 3295931587706880000, 98555678764852838400, 2959750227906986803200

Offset: 0

Paul Barry, Feb 14 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A060150.

[2^(n-1)*Binomial(2*n-1,n-1)^2: n in [0..20]]; // Vincenzo Librandi, Nov 17 2011

Join[{0},Table[2^(n-1) Binomial[2n-1,n-1]^2,{n,20}]] (* Harvey P. Dale, Dec 29 2023 *)

a(n)=binomial(2*n-1,n-1)^2<<(n-1) \\ Charles R Greathouse IV, Oct 23 2023

G.f.: 1+(K(32x)-1)/4 where K(k)=Elliptic_F(pi/2,k) is the complete Elliptic integral of the first kind;
e.g.f.: BesselI(0, 2*sqrt(2)x)*BesselI(1, 2*sqrt(2)x)/sqrt(2);
a(n) = 2^(n+1)*(binomial(2n,n)/4)^2 - 0^n/8.
Conjecture: n^2*a(n) - (2*n-1)^2*a(n-1) = 0. - R. J. Mathar, Nov 16 2011

A277584 a(n) = binomial(3n-1, n-1)^2.

Original entry on oeis.org

0, 1, 25, 784, 27225, 1002001, 38291344, 1502337600, 60101954649, 2440703175625, 100300325150025, 4161829109817600, 174077451630810000, 7330421677037621904, 310467090932230849600, 13214837914326197526784, 564927069263895118093401

Offset: 0

Seiichi Manyama, Oct 22 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..605

Cf. A025174, A060150, A188662.

[Binomial(3*n-1, n-1)^2: n in [0..20]]; // Vincenzo Librandi, Oct 23 2016

Table[Boole[n > 0] Binomial[3 n - 1, n - 1]^2, {n, 0, 16}] (* Michael De Vlieger, Oct 26 2016 *)

a(n) = binomial(3*n-1, n-1)^2; \\ Michel Marcus, Oct 22 2016

a(n) = A025174(n)^2.
a(n) = A188662(n)/9 for n > 0.
Let the number of multisets of length k on n symbols be denoted by ((n, k)) = binomial(n+k-1, k).
a(n) = (Sum_{k=0..n} binomial(n, k)^2 * ((2*n, 2*n - k)))/5 for n > 0.

A337870 The number of random walks on the simple square lattice that start at the origin (0,0) and pass through (1,0) after 2n+1 steps before having returned to the origin.

Original entry on oeis.org

1, 2, 16, 166, 1934, 24076, 312906, 4191822, 57433950, 800740450, 11319707546, 161841539812, 2335765140994, 33979681977530, 497696233487200, 7332776490675630, 108595186409772174, 1615573668169487898, 24132221328987714066

Offset: 0

R. J. Mathar, Sep 27 2020

The number of walks that take one of the four directions U, D, R, L which arrive at (1,0) is zero if the number of steps is even. For odd number of steps we count the walks that start at (0,0) pass through any set of points that are not {(0,0),(1,0)} and arrive at (1,0).

The ordinary generating function is a mix of inverses of sums and differences of the hypergeometric generating functions in A002894 and A060150. See Maple.

Cf. A002894, A060150, A275912, A337869.

g002894 := hypergeom([1/2,1/2],[1],16*x^2) ;
g060150 := x*hypergeom([1,3/2,3/2],[2,2],16*x^2) ;
1/2/(g002894-g060150)-1/2/(g002894+g060150) ;
taylor(%,x=0,40);
L := gfun[seriestolist](%) ; # includes zeros of even steps

cp's OEIS Frontend

A254686 Number of ways to put n red and n blue balls into n indistinguishable boxes.

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A337902 The number of walks of length 2n+1 on the square lattice that start from the origin (0,0) and end at the vertex (2,1).

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A337869 The number of random walks on the simple square lattice that return to the origin (0,0) after 2n steps and do not pass through (0,0) or (1,0) at intermediate steps.

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A337901 The number of walks of length 2n+1 on the square lattice that start from the origin (0,0) and end at the vertex (3,0).

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A380241 Array read by antidiagonals: T(n,k) is the number of rooted (2k)-regular planar maps with n vertices, n >= 0, k >= 0.

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PARI

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A378060 a(n) = binomial(n, floor((n-1)/2))^2.

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Julia

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A378070 a(n) = binomial(n - 1, ceiling(n/2)) * binomial(n - 1, ceiling(n/2) - 1).

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A116421 a(n) = 2^(n-1)*binomial(2n-1,n-1)^2.

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Magma

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A277584 a(n) = binomial(3n-1, n-1)^2.