Feng Jishe - cp's OEIS frontend

User: Feng Jishe

Feng Jishe has authored 5 sequences.

A335857 a(n) is the determinant of the n X n Hankel matrix A with A(i,j) = A000108(i+j+6) for 0<=i,j<=n-1.

1, 132, 4719, 81796, 884884, 6852768, 41314284, 204951252, 869562265, 3245256300, 10880587575, 33309352440, 94307358288, 249485071616, 621856804272, 1470540624696, 3318218562009, 7179339254516, 14955909351383, 30104651175324, 58733021049780, 111358254207200

Offset: 0

Feng Jishe, Oct 12 2020

Starting with 132 = A000108(6), the sequence is the Hankel transform of the Catalan numbers shifted to remove the first 6 terms.

			a(1) = 132 because 132 is the determinant of the 1 X 1 matrix [132].
a(2) = 4719 because 4719 is the determinant of the matrix
    [ 132  429 ]
    [ 429 1430 ].
a(3) = 81796 because 81796 is the determinant of the matrix
    [  132,  429,  1430 ]
    [  429, 1430,  4862 ]
    [ 1430, 4862, 16796 ].
G.f. = 1 + 132*x + 4719*x^2 + 81796*x^3 + 884884*x^4 + ... - _Michael Somos_, Jun 27 2023

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jishe Feng, The explicit formula of Hankel determinant with Catalan elements, arXiv:2010.06586 [math.GM], 2020.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 24.
Index entries for linear recurrences with constant coefficients, signature (16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1).

Hankel transforms of the Catalan numbers shifted by k for k=0..5 are A000012, A000012, A000027, A000330, A006858, A091962.

Cf. A000108 (Catalan numbers).

Table[Binomial[n+5, 5]*Binomial[2*n+7, 3]*Binomial[2*n+9, 7]/1260, {n,0,30}] (* G. C. Greubel, Dec 17 2021 *)
a[ n_] := Binomial[n+5, 4]*Binomial[2*n+7, 3]*Binomial[2*n+9, 8]/1575; (* Michael Somos, Jun 27 2023 *)

H(seq)={my(n=(#seq+1)\2); matdet(matrix(n,n,i,j,seq[i+j-1]))}
a(n, k=6)={H(vector(2*n, i, my(t=i+k-1); binomial(2*t,t)/(t+1)))} \\ Andrew Howroyd, Nov 26 2020

{a(n) = prod(k=1, 5, (n+k)^min(6-k, k)) * prod(k=1, 4, (2*n+2*k+1)^min(5-k, k))/285768000}; /* Michael Somos, Jun 27 2023 */

[binomial(n+5, 5)*binomial(2*n+7, 3)*binomial(2*n+9, 7)/1260 for n in (0..30)] # G. C. Greubel, Dec 17 2021

a(n) = 2^5/(5 * 7!* 9!)*(n + 1)*(n + 2)^2*(n + 3)^3 *(n + 4)^2*(n + 5)*(2*n + 3)*(2*n + 5)^2*(2*n + 7)^2*(2*n + 9).
From G. C. Greubel, Dec 17 2021: (Start)
a(n) = binomial(n+5, 5)*binomial(2*n+7, 3)*binomial(2*n+9, 7)/1260.
G.f.: (1 + 116*x + 2727*x^2 + 21572*x^3 + 70328*x^4 + 103376*x^5 + 70328*x^6 + 21572*x^7 + 2727*x^8 + 116*x^9 + x^10)/(1 - x)^16. (End)
a(n) = -a(-6-n) for all n in Z. - Michael Somos, Jun 27 2023
a(n) ~ n^15/4465125. - Stefano Spezia, Dec 09 2023

A296619 The number of nonnegative walks of n steps with step sizes 1 and 2, starting at 0 and ending at 2.

Original entry on oeis.org

0, 1, 1, 6, 13, 52, 152, 550, 1813, 6453, 22427, 80330, 286895, 1038931, 3772801, 13807294, 50726893, 187332517, 694364517, 2583714636, 9644852364, 36115537269, 135607526865, 510496492338, 1926284451923, 7284476707597, 27602839227883, 104791979218326

Offset: 0

Feng Jishe, Dec 17 2017

a(n) is the number of 2-D walks with n steps of type {(1,-2), (1,-1), (1,1), or (1,2)} starting at (0,0), ending at (n,2), and not dropping below the x-axis.

The sequence corresponds to element (1,3) of the matrix B(n)^n (see Maple script). Furthermore, element (1,1) of the matrix is A187430, the element (1,2) of these matrix is A055113.

			There are 6 walks of length 3:
        __
       |  |         __
     __|  |_     __|  |_     __    _
    |           |           |  |__|
   _|          _|          _|
    2+2-2=2     2+1-1=2     2-1+1=2
                    __
     __    _       |  |_           _
    |  |  |      __|         __   |
   _|  |__|    _|          _|  |__|
    2-2+2=2     1+2-1=2     1-1+2=2

Robert Israel, Table of n, a(n) for n = 0..1667

Cf. A055113, A185286, A187430.

B := n -> LinearAlgebra:-ToeplitzMatrix([0,1,1, seq(0, k=0..n-2)], symmetric):
seq((B(n)^n)(1, 3), n=0..27);
# alternative:
T:= proc(n,k) option remember;
  if k < 0 or k > 2*n then return 0 fi;
  procname(n-1,k-2)+procname(n-1,k-1)+procname(n-1,k+1)+procname(n-1,k+2)
end proc:
T(0,0):= 1:
seq(T(n,2),n=0..40); # Robert Israel, Dec 19 2017

b[n_] := ToeplitzMatrix[Join[{0,1,1}, ConstantArray[0,n-1]]];
Prepend[Table[MatrixPower[b[n],n][[1,3]], {n,20}], 0]
(* Andrey Zabolotskiy, Dec 19 2017 *)

Next(v)={vector(#v+2, i, if(i<3||i>#v-2, 0, v[i-2]+v[i-1]+v[i+1]+v[i+2]))}
my(v=vector(7,i,i==3)); for(n=1, 50, print1(v[5],", "); v=Next(v)) \\ Andrew Howroyd, Dec 18 2017

a(n) = A185286(n,2). - Robert Israel, Dec 19 2017

A277248 Number of planar walks starting at (1,1), ending at (3n,0), remaining in the first quadrant and using steps (-1,2) and (2,-1).

Original entry on oeis.org

1, 2, 6, 24, 108, 528, 2724, 14616, 80760, 456552, 2628504, 15360216, 90879096, 543336912, 3277586136, 19924733088, 121943223576, 750756116376, 4646484480552, 28892787031008, 180420486241776, 1130930538186360, 7113550964713848, 44885329202906448

Offset: 1

Feng Jishe, Oct 06 2016

Alois P. Heinz, Table of n, a(n) for n = 1..1000
M. Bousquet-Mélou, M. Petkovsek, Walks confined in a quadrant are not always D-finite, Theoretical Computer Science, 307(2003): 257-276.
Ira M. Gessel, A probabilistic method for lattice path enumeration, Journal of statistical planning and inference, 14 (1986), 49-58.

Cf. A048116.

b:= proc(l) option remember; `if`(l=[1$2], 1, add((p->
      `if`(p[1]<0, 0, b(p)))(sort((l-x))), x=[[-1, 2], [2, -1]]))
    end:
a:= n-> b([0,3*n]):
seq(a(n), n=1..30);  # Alois P. Heinz, Oct 06 2016

b[l_List] := b[l] = If[l == {1, 1}, 1, Sum[Function[p, If[p[[1]]<0, 0, b[p]]][Sort[l-x]], {x, {{-1, 2}, {2, -1}}}]]; a[n_] := b[{0, 3n}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Dec 04 2016 after Alois P. Heinz *)

a(n) ~ c * (27/4)^n / n^(3/2), where c = 0.06045583689606517807688682344735167414726208387456561322459238109992522838... . - Vaclav Kotesovec, Oct 07 2016

More terms from Alois P. Heinz, Oct 06 2016

A270174 a(n) is the number of different ways to seat a set of n married male-female couples at a straight table so that men and women alternate and every man is separated by at least two men from his wife.

Original entry on oeis.org

0, 0, 0, 0, 240, 8640, 584640, 40239360, 3493808640, 364941158400, 45683021260800, 6754660222464000, 1166167699041945600, 232618987254682828800, 53114643986227439616000, 13768242163527512973312000, 4021980517038414919532544000, 1315337131173516220415213568000

Offset: 1

Feng Jishe, Mar 12 2016

nonn

We assume that the chairs are uniform and indistinguishable.

First we arrange the women in alternating seats, in 2*n! ways. Second, we find the number, G_{n} say, of ways of arranging men in the remaining seats such that every husband cannot sit at the left or right next 1, 2, ..., h male's seats from his wife. Note that here h = 2. We give the board B4, where X denotes the seat cannot be set at, where there are h X's in first column, and h+1 X's in first row, ..., 2h X's in the h column, ..., other entries are 1's. Thus the number of different ways to seat a set of n married male-female couples at a straight table is a_{n}=2*n!*G_{n}.

Feng Jishe, The board B4
D. Zeilberger, Automatic Enumeration of Generalized Ménage Numbers
D. Zeilberger, Automatic Enumeration of Generalized Menage Numbers, arXiv preprint arXiv:1401.1089 [math.CO], 2014.

Cf. A267060, A292574.

a(n) = 2*n! * A292574(n). - Andrew Howroyd, Sep 19 2017

a(11)-a(18) from Andrew Howroyd, Sep 19 2017

A267060 a(n) = number of different ways to seat a set of n married male-female couples at a round table so that men and women alternate and every man is separated by at least d = 2 men from his wife.

Original entry on oeis.org

0, 0, 0, 0, 24, 240, 22320, 1330560, 112210560, 11183235840, 1340192044800, 189443216793600, 31267307962598400, 5964702729085900800, 1303453560329957836800, 323680816052170536960000, 90679832709074132299776000, 28473630606612014817337344000

Offset: 1

Feng Jishe, Jan 09 2016

nonn

We assume that the chairs are uniform and indistinguishable.
First we arrange the females in alternating seats by circular permutation, there are (n-1)! ways. Secondly, we evaluate the number F_{n}, ways of arranging males in the remaining seats as mentioned in the definition above.
By the principle of inclusion-exclusion and theory of rook polynomial Vl, we obtain that a_{n} = (n-1)!*F_{n}, F_{n} = sum(-1)^{k}*r_{k}(B3)*(n-k)! where r_{k}(B3) is the number of ways of putting k non-taking rooks on positions 1's of B3, and the rook polynomials are R_{B3}(x) = sum r_{k}(B3)*x^{k}.
Also F_{n} = per(B3), here per(B3) denotes the permanent of matrix (board) B3, but it is very difficult problem to evaluate the value, per(B3).

			For d=1, the sequence a_{n} is the classical menage sequence A094047.
For d=2 (the current sequence), the F(n)s are 0, 0, 0, 0, 1, 2, 31, 264, 2783, 30818, 369321, ... which is A004307(n) then the sequence a_{n} is 0, 0, 0, 0, 24, 240, 22320, 1330560, 112210560, 11183235840, 1340192044800,...
For d=3, the F(n)s are 0, 0, 0, 0, 0, 0, 1, 2, 78, 888, 13909, ... which is A184965, and a(n) = (n-1)!*A184965(n).

G. Polya, Aufgabe 424, Arch. Math. Phys. (3) 20 (1913) 271.
John Riordan. The enumeration of permutations with three-ply staircase restrictions.

Alois P. Heinz, Table of n, a(n) for n = 1..253
E. Rodney Canfield and Nicholas C. Wormald, Menage numbers, bijections and P-recursiveness, Discrete Mathematics, 63 (2--3)(1987): 117--129.
Bruno Codenotti, Giovanni Resta, On the permanent of certain circulant matrices, in Algebraic Combinatorics and Computer Science. 513-532. 2001.
Feng Jishe, Illustration
Yiting Li, Ménage Numbers and Ménage Permutations, Journal of Integer Sequences, Vol. 18 (2015), #15.6.8.
M. Marcus and H. Mint, On the relation between the determinant and the permanent, Illinois J. Math. 5 (1961): 376-381.
N. Metropolis, M. L. Stein, P. R. Stein, Permanents of cyclic (0,1) matrices, J. Combin. Theory, 7 (1969), 291-321.
Giovanni Sburlati, On the values of permanents of (0,1) circulant matrices with three ones per row, Linear Algebra and its Applications. 408 (2005) 284--297.
Vladimir Shevelev, Peter J.C. Moses, The menage problem with a fixed couple, arXiv:1101.5321 [math.CO], 2011-2015.
L. G. Valiant, The complexity of computing the permanent, Theoret. Comput. Sci. 8 (1979): 189-201.
Vijay V. Vazirani, Milhalis Yannakakis, Pfaffian orientations, 0-1 permanents, and even cycles in directed graphs, Discrete Applied Mathematics, 25(1989): 179-190.
M. Wyman and L. Moser, On the problème des ménages, Canad. J. Math., 10 (1958), 468-480.

Cf. A004307, A094047, A184965.

b[n_, n0_] := Permanent[Table[If[(0 <= j - i && j - i < n - n0) || j - i < -n0, 1, 0], {i, 1, n}, {j, 1, n}]];
A004307[n_] := b[n, 4];
a[n_] := (n - 1)!*A004307[n];
Array[a, 18] (* Jean-François Alcover, Oct 08 2017 *)

a(n) = (n-1)! * A004307(n). - Andrew Howroyd, Sep 19 2017

a(12)-a(18) from Andrew Howroyd, Sep 19 2017

cp's OEIS Frontend

User: Feng Jishe

A335857 a(n) is the determinant of the n X n Hankel matrix A with A(i,j) = A000108(i+j+6) for 0<=i,j<=n-1.

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A296619 The number of nonnegative walks of n steps with step sizes 1 and 2, starting at 0 and ending at 2.

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A277248 Number of planar walks starting at (1,1), ending at (3n,0), remaining in the first quadrant and using steps (-1,2) and (2,-1).

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A270174 a(n) is the number of different ways to seat a set of n married male-female couples at a straight table so that men and women alternate and every man is separated by at least two men from his wife.

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A267060 a(n) = number of different ways to seat a set of n married male-female couples at a round table so that men and women alternate and every man is separated by at least d = 2 men from his wife.

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