A050447 -id:A050447 - cp's OEIS frontend

A006360 Antichains (or order ideals) in the poset 223n or size of the distributive lattice J(223n).

1, 50, 887, 8790, 59542, 307960, 1301610, 4701698, 14975675, 43025762, 113414717, 277904900, 639562508, 1393844960, 2896063220, 5768600412, 11066514565, 20526933442, 36936277875, 64660182026, 110394412610

Offset: 0

N. J. A. Sloane

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.
Manfred Goebel, Rewriting Techniques and Degree Bounds for Higher Order Symmetric Polynomials, Applicable Algebra in Engineering, Communication and Computing (AAECC), Volume 9, Issue 6 (1999), 559-573.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.
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. 9.
Index entries for sequences related to posets.

Cf. A000217, A000330, A050446, A050447, A006356, A006357, A006358, A006359, A000372, A056932, A006361, A006362, A056933, A056934, A056935, A056936, A056937.

Empirical G.f.: (x+1)*(x^6+36*x^5+279*x^4+594*x^3+279*x^2+36*x+1)/(1-x)^13. - Colin Barker, May 29 2012

More terms from Mitch Harris, Jul 16 2000

A079502 Triangle T(n,k) read by rows; related to number of preorders.

Original entry on oeis.org

1, 1, 2, 1, 5, 5, 1, 10, 24, 16, 1, 18, 79, 122, 61, 1, 31, 223, 602, 680, 272, 1, 52, 579, 2439, 4682, 4155, 1385, 1, 86, 1432, 8856, 25740, 38072, 27776, 7936, 1, 141, 3434, 30030, 124146, 272416, 326570, 202084, 50521, 1, 230, 8071, 97332

Offset: 0

N. J. A. Sloane, Jan 21 2003

			Triangle T(n,k) begins:
  1;
  1,  2;
  1,  5,    5;
  1, 10,   24,  16;
  1, 18,   79,  122,    61;
  1, 31,  223,  602,   680,   272;
  1, 52,  579, 2439,  4682,  4155,  1385;
  1, 86, 1432, 8856, 25740, 38072, 27776, 7936;

Sean A. Irvine, Table of n, a(n) for n = 0..999
Germain Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30. (The numbers u_r^n on page 20.)
Germain Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30. (Annotated scanned copy)
Igor Pak, Boris Shapiro, Ilya Smirnov, and Ken-ichi Yoshida, Hilbert-Kunz multiplicity of quadrics via the Ehrhart theory, Stockholm Univ. (Sweden, 2025). See pp. 5, 10.

Diagonals give A000111, A006326, A006327, A006328. Cf. A050447.

t[n_, m_] := t[n, m] = If[m == 0, 1, t[n, m - 1] + Sum[t[2 k, m - 1] t[n - 1 - 2 k, m], {k, 0, (n - 1)/2}]]; Map[Function[s, Rest@ Reverse@ Map[Abs@ Fold[#2 - #1 &, Reverse@ Take[s, #]] &, Range@ Length@ s]]@ Reverse@ Map[First, NestList[Differences@ # &, {First@ #}~Join~Differences@ #, Length@ # - 2]] &, Table[t[n, k], {n, 2, 11}, {k, 0, n}]] (* Michael De Vlieger, Mar 13 2017, after Jean-François Alcover at A050447 *)

From Sean A. Irvine, Mar 12 2017: (Start)
A079502 can be constructed one row at a time from the corresponding row of A050447. For row n, construct up to the n-th difference sequence of row n in A050447, retaining the first element of each difference sequence. Row n of A079502 is then constructed backwards (i.e., starting with A079502(n,n) and computing down to A079502(n,2)) from the first element of the n-th difference sequence, then successively subtracting the first element of the previous difference sequences. More precisely, let R_n denote the n-th row of A050447 augmented with R_n(1) = 0, and R_n^(d) the d-th difference of that row, such that R_n^(0)(m) = R_n(m) and R_n^(k)(m) = R_n^(k-1)(m+1) - R_n^(k-1)(m). Row n of A079502 is then T(n,n) = R_n^(n)(0) and for m < n, T(n,m) = R_n^(n)(0) - T(n,m+1).
For example, starting with row 4 of A050447: [0], 1, 8, 31, 85, 190, 371, ..., we construct up to order 4 difference sequences: first-differences 1, 7, 23, 54, 105, 181, ...; second-differences 6, 16, 31, 51, 76, ...; third-differences 10, 15, 20, 25, ...; fourth-differences 5, 5, 5, ... (constant). Only the first elements of these difference sequences are needed. Thus T(4,4) = 5, T(4,3) = 10 - 5 = 5, T(4,2) = 6 - (10 - 5) = 1, T(4,1) = 1 - (6 - (10 - 5)) = 0. (End)

More terms from Sean A. Irvine, Mar 12 2017

A276312 Number of up-down sequences of length n and values in {1,2,...,n}.

Original entry on oeis.org

1, 1, 1, 5, 31, 246, 2353, 26585, 345775, 5094220, 83833256, 1524414737, 30353430420, 656851828075, 15350023574061, 385261255931365, 10335781852020335, 295166535640444376, 8939894824857438940, 286234265613041061128, 9659753724363828753408

Offset: 0

Alois P. Heinz, Aug 29 2016

nonn

			a(0) = 1: the empty sequence.
a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 5: 121, 131, 132, 231, 232.
a(4) = 31: 1212, 1213, 1214, 1312, 1313, 1314, 1323, 1324, 1412, 1413, 1414, 1423, 1424, 1434, 2312, 2313, 2314, 2323, 2324, 2412, 2413, 2414, 2423, 2424, 2434, 3412, 3413, 3414, 3423, 3424, 3434.

Alois P. Heinz, Table of n, a(n) for n = 0..413

A diagonal of A050446, A050447.

Cf. A276313.

b:= proc(n, k, t) option remember; `if`(n=0, 1,
      add(b(n-1, k, k-j), j=1..t-1))
    end:
a:= n-> b(n, n+1$2):
seq(a(n), n=0..25);

b[n_, k_, t_] := b[n, k, t] = If[n==0, 1, Sum[b[n-1, k, k-j], {j, 1, t-1}]];
a[n_] := b[n, n+1, n+1];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 29 2020, after Alois P. Heinz *)

a(n) ~ exp(-1/2) * 2^(n+2) * n^n / Pi^(n+1). - Vaclav Kotesovec, Aug 30 2016

A276313 Number of weak up-down sequences of length n and values in {1,2,...,n}.

Original entry on oeis.org

1, 1, 3, 14, 85, 671, 6405, 72302, 940005, 13846117, 227837533, 4142793511, 82488063476, 1785049505682, 41715243815059, 1046997553798894, 28089178205661221, 802173732190546289, 24296253228394108980, 777918130180655893150, 26253270588637259772768

Offset: 0

Alois P. Heinz, Aug 29 2016

nonn

			a(0) = 1: the empty sequence.
a(1) = 1: 1.
a(2) = 3: 11, 12, 22.
a(3) = 14: 111, 121, 122, 131, 132, 133, 221, 222, 231, 232, 233, 331, 332, 333.
a(4) = 85: 1111, 1112, 1113, 1114, 1211, ..., 4423, 4424, 4433, 4434, 4444.

Alois P. Heinz, Table of n, a(n) for n = 0..413

A diagonal of A050446, A050447.

Cf. A276312.

b:= proc(n, k, t) option remember; `if`(n=0, 1,
      add(b(n-1, k, k-j), j=1..t))
    end:
a:= n-> b(n, n+1, n):
seq(a(n), n=0..25);

b[n_, k_, t_] := b[n, k, t] = If[n==0, 1, Sum[b[n-1, k, k-j], {j, 1, t}]];
a[n_] := b[n, n+1, n];
Table[a[n], {n, 0, 25}](* Jean-François Alcover, May 18 2017, translated from Maple *)

a(n) ~ exp(1/2) * 2^(n+2) * n^n / Pi^(n+1). - Vaclav Kotesovec, Aug 30 2016

A108582 n appears n^3 times.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Jonathan Vos Post, Jul 25 2005

From Jonathan Vos Post, Mar 18 2006: (Start)
The key to this sequence is: 1^3 + 2^3 + 3^3 + ... + n^3 = (1+2+3+...+n)^2.
Since the last occurrence of n comes one before the first occurrence of n+1 and the former is at Sum_{i=0..n} i^3 = A000537(n) = (A000217(n))^2 = (n*(n+1)/2)^2 = (C(n+1,2))^2, have a(A000537(n)) = a((A000217(n))^2) = n and thus a(1+A000537(n)) = a(1+(A000217(n))^2) = n+1.
The current sequence is, loosely, the inverse function of the square of the triangular number sequence. (End)

Boris Putievskiy, Table of n, a(n) for n = 1..8281
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.

Cf. A000027, A000578, A002024, A072649, A074279, A000217.

Cf. A000330, A000537, A006331, A050446, A050447, A006003, A005900.

Flatten @ Table[ Table[k, {k^3}], {k, 5}] (* Giovanni Resta, Jun 17 2016 *)
a[n_]:=Ceiling[1/2 (Sqrt[8 Sqrt[n]+1]-1)]
Nmax=225; Table[a[n],{n,1,Nmax}] (* Boris Putievskiy, Jun 19 2024 *)

from sympy import integer_nthroot
def A108582(n): return (m:=integer_nthroot(k:=n<<2,4)[0])+(k>(m*(m+1))**2) # Chai Wah Wu, Nov 04 2024

a(n) = ceiling((1/2)*(sqrt(8*sqrt(n) + 1) - 1)). - Boris Putievskiy, Jun 19 2024
From Chai Wah Wu, Nov 04 2024: (Start)
a(n) = m+1 if n>(m(m+1))^2/4 and a(n) = m otherwise where m = floor((4n)^(1/4)).
More generally, for a sequence a_k(n) where n appears n^(k-1) times, a_k(n) = m+1 if n > Sum_{i=1..m} i^(k-1) and a_k(n) = m otherwise where m = floor((kn)^(1/k)).
Note that Sum_{i=1..m} i^(k-1) can be written as a k-th order polynomial of m using Faulhaber's formula. (End)

Two missing terms from Giovanni Resta, Jun 17 2016

A205492 Expansion of (1-x^2-x^3-x^4+x^5)/((1-x)^3(1-x-x^2)^2(1-2*x-x^2+x^3)).

Original entry on oeis.org

1, 7, 31, 109, 334, 937, 2475, 6267, 15393, 36976, 87369, 203915, 471546, 1082849, 2473535, 5627684, 12765052, 28887838, 65260270, 147233926, 331842395, 747355066, 1682185342, 3784718431, 8512408455, 19141037360, 43032743620

Offset: 0

L. Edson Jeffery, Jan 28 2012

nonn

See array A205497 regarding association of this sequence with generating functions for the rows of the array form of A050446.

G. C. Greubel, Table of n, a(n) for n = 0..1000
L. E. Jeffery, Unit-primitive matrices
Index entries for linear recurrences with constant coefficients, signature (7,-17,12,15,-26,3,13,-5,-2,1).

Cf. A050446, A050447, A205497.

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x^2-x^3-x^4+x^5)/((1-x)^3*(1-x-x^2)^2*(1-2*x-x^2+ x^3)) )); // G. C. Greubel, Jan 04 2020

seq(coeff(series((1-x^2-x^3-x^4+x^5)/((1-x)^3*(1-x-x^2)^2*(1-2*x-x^2+ x^3)), x, n+1), x, n), n = 0..30); # G. C. Greubel, Jan 04 2020

LinearRecurrence[{7,-17,12,15,-26,3,13,-5,-2,1},{1,7,31,109,334,937,2475,6267, 15393,36976},30] (* Harvey P. Dale, Mar 26 2013 *)
CoefficientList[Series[(1-x^2-x^3-x^4+x^5)/((1-x)^3*(1-x-x^2)^2*(1-2*x-x^2+ x^3)), {x,0,30}], x] (* G. C. Greubel, Jan 04 2020 *)

my(x='x+O('x^30)); Vec((1-x^2-x^3-x^4+x^5)/((1-x)^3*(1-x-x^2)^2*(1-2*x-x^2+ x^3))) \\ G. C. Greubel, Jan 04 2020

def A205492_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x^2-x^3-x^4+x^5)/((1-x)^3*(1-x-x^2)^2*(1-2*x-x^2+x^3)) ).list()
A205492_list(30) # G. C. Greubel, Jan 04 2020

a(n) = 7*a(n-1) - 17*a(n-2) + 12*a(n-3) + 15*a(n-4) - 26*a(n-5) + 3*a(n-6) + 13*a(n-7) - 5*a(n-8) - 2*a(n-9) + a(n-10), n>9, {a(m)} = {1, 7, 31, 109, 334, 937, 2475, 6267, 15393, 36976}, m=0,...,9.
CONJECTURE 1. a(n) = M_{n,2} = M_{2,n}, where M = A205497.
CONJECTURE 2. lim_{n->oo} a(n+1)/a(n) = (2*cos(Pi/7))^2-1 = A116425-1 = spectral radius of the 3 X 3 unit-primitive matrix (see [Jeffery]) A_{7,2} = [0,0,1; 0,1,1; 1,1,1].

A373424 Array read by ascending antidiagonals: T(n, k) = [x^k] cf(n) where cf(n) is the continued fraction (-1)^n/(~x - 1/(~x - ... 1/(~x - 1)))...) and where '~' is '-' if n is even, and '+' if n is odd, and x appears n times in the expression.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 6, 5, 1, 0, 1, 5, 10, 14, 8, 1, 0, 1, 6, 15, 30, 31, 13, 1, 0, 1, 7, 21, 55, 85, 70, 21, 1, 0, 1, 8, 28, 91, 190, 246, 157, 34, 1, 0, 1, 9, 36, 140, 371, 671, 707, 353, 55, 1, 0, 1, 10, 45, 204, 658, 1547, 2353, 2037, 793, 89, 1, 0

Offset: 0

Peter Luschny, Jun 09 2024

A variant of both A050446 and A050447 which are the main entries. Differs in indexing and adds a first row to the array resp. a diagonal to the triangle.

			Generating functions of the rows:
   gf0 =  1;
   gf1 = -1/( x-1);
   gf2 =  1/(-x-1/(-x-1));
   gf3 = -1/( x-1/( x-1/( x-1)));
   gf4 =  1/(-x-1/(-x-1/(-x-1/(-x-1))));
   gf5 = -1/( x-1/( x-1/( x-1/( x-1/( x-1)))));
   gf6 =  1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1))))));
   ...
Array A(n, k) starts:
  [0] 1, 0,  0,  0,   0,    0,    0,     0,      0,      0, ...  A000007
  [1] 1, 1,  1,  1,   1,    1,    1,     1,      1,      1, ...  A000012
  [2] 1, 2,  3,  5,   8,   13,   21,    34,     55,     89, ...  A000045
  [3] 1, 3,  6, 14,  31,   70,  157,   353,    793,   1782, ...  A006356
  [4] 1, 4, 10, 30,  85,  246,  707,  2037,   5864,  16886, ...  A006357
  [5] 1, 5, 15, 55, 190,  671, 2353,  8272,  29056, 102091, ...  A006358
  [6] 1, 6, 21, 91, 371, 1547, 6405, 26585, 110254, 457379, ...  A006359
   A000027,A000330,   A085461,     A244881, ...
       A000217, A006322,    A108675, ...
.
Triangle T(n, k) = A(n - k, k) starts:
  [0] 1;
  [1] 1,  0;
  [2] 1,  1,  0;
  [3] 1,  2,  1,  0;
  [4] 1,  3,  3,  1,  0;
  [5] 1,  4,  6,  5,  1,  0;
  [6] 1,  5, 10, 14,  8,  1, 0;

T. Kyle Petersen and Yan Zhuang, Zig-zag Eulerian polynomials, arXiv:2403.07181 [math.CO], 2024. (Table 3)

Cf. A050446, A050447, A276313 (main diagonal), A373353 (row sums of triangle).

Cf. A373423.

row := proc(n, len) local x, a, j, ser; if irem(n, 2) = 1 then
a :=  x - 1; for j from 1 to n do a :=  x - 1 / a od: a :=  a - x; else
a := -x - 1; for j from 1 to n do a := -x - 1 / a od: a := -a - x;
fi; ser := series(a, x, len + 2); seq(coeff(ser, x, j), j = 0..len) end:
A := (n, k) -> row(n, 12)[k+1]:      # array form
T := (n, k) -> row(n - k, k+1)[k+1]: # triangular form

def Arow(n, len):
    R. = PowerSeriesRing(ZZ, len)
    if n == 0: return [1] + [0]*(len - 1)
    x = -x if n % 2 else x
    a = x + 1
    for _ in range(n):
        a = x - 1 / a
    a = x - a if n % 2 else a - x
    return a.list()
for n in range(7): print(Arow(n, 10))

A205493 Third row or column of table A205497.

Original entry on oeis.org

1, 14, 109, 623, 2951, 12331, 47191, 169416, 579889, 1914226, 6144668, 19298724, 59579803, 181448918, 546629054, 1632497850, 4841448042, 14277423006, 41912838982, 122587133760, 357476552161, 1039922075888, 3019280091491, 8752184436454, 25337900299765

Offset: 0

L. Edson Jeffery, Jan 28 2012

nonn

See A205497 regarding association of this sequence with generating functions for the rows of the tabular form of A050446.

L. E. Jeffery, Unit-primitive matrices

Cf. A205497, A050446, A050447.

Python
```
# See program in A205497.
```

Conjecture 1. a(n) = M_{n,3} = M_{3,n}, where M = A205497.

Conjecture 2. Let w=2*cos(Pi/9). Then lim_{n->oo} a(n+1)/a(n) = w^3-2*w = spectral radius of the 4 X 4 unit-primitive matrix (see [Jeffery]) A_{9,3} = [0,0,0,1; 0,0,1,1; 0,1,1,1; 1,1,1,1].

a(24) and changed title from Hugo Pfoertner, Jan 05 2020

A205494 Conjectured row or column n=4 of array A205497.

Original entry on oeis.org

1, 26, 334, 2951, 20641, 123216, 656683, 3217526, 14786816, 64657546, 271838823, 1107586989, 4399926007, 17122243560, 65514790830, 247212893755, 922136438698, 3406871213836, 12486569116765, 45459575562313, 164578100859837, 593025025473647, 2128399709975819, 7613495897772440

Offset: 0

L. Edson Jeffery, Jan 28 2012

nonn

The denominator of the generating function for this sequence is a polynomial of degree 35. Terms corresponding to n=0,...,23 are shown above, with those for n=24,...,40 as follows: {27157723973468595, 96643368020414337, 343226612286408932, 1216901732483780905, 4308339945395597755, 15234940157670046379, 53818220864065451564, 189952299613455045068, 669953408386151161398, 2361449534293944339096, 8319329987059336296021, 29296032314800671782284, 103126374236214419873734, 362907786820798388773987, 1276761054260676178577043, 4490840947292979020061377, 15793032895427304036405557}.

See A205497 regarding association of this sequence with generating functions for the rows of the tabular form of A050446.

L. E. Jeffery, Unit-primitive matrices

Cf. A205497, A050446, A050447.

G.f.: (1+4*x-31*x^2 - 67*x^3 + 348*x^4 + 418*x^5 - 1893*x^6 - 1084*x^7 + 4326*x^8 + 4295*x^9 - 7680*x^10 - 9172*x^11 + 9104*x^12 + 11627*x^13 - 5483*x^14 - 10773*x^15 + 1108*x^16 + 7255*x^17 + 315*x^18 - 3085*x^19 - 228*x^20 + 669*x^21 + 102*x^22 - 23*x^23 - 45*x^24 - 16*x^25 + 11*x^26 + 2*x^27 - x^28) / ((1-x)^5 * (1-x-x^2)^4 * (1-2*x-x^2+x^3)^3 * (1-2*x-3*x^2+x^3+x^4)^2 * (1-3*x-3*x^2+4*x^3+x^4-x^5)).
CONJECTURE 1. a(n) = M_{n,4} = M_{4,n}, where M = A205497.
CONJECTURE 2. Let w=2*cos(Pi/11). Then lim_{n->oo} a(n+1)/a(n) = w^4-3*w^2+1 = spectral radius of the 5 X 5 unit-primitive matrix (see [Jeffery]) A_{11,4} = [0,0,0,0,1; 0,0,0,1,1; 0,0,1,1,1; 0,1,1,1,1; 1,1,1,1,1].

A205495 Convolution related to array A205497 and to generating functions for the rows of the array form of A050446.

Original entry on oeis.org

1, 46, 937, 12331, 123216, 1019051, 7349140, 47816612, 287357460, 1622135139, 8709442871, 44899559053, 223883501478, 1086005140508, 5148332487873, 23940669359515, 109535136537197, 494307574790201, 2204762394907238, 9736270202183689, 42629974672006973

Offset: 0

L. Edson Jeffery, Jan 28 2012

nonn

The denominator of the generating function for this sequence is a polynomial of degree 56. Terms corresponding to n=0,...,20 are shown above, with those for n=21,...,60 as follows: {185291835954412064, 800317930217099771, 3438057983187970745, 14700487950597800766, 62602970565114993286, 265668524077091893747, 1124012759249695584332, 4743119424920236606646, 19969635838069446154607, 83911303727287364502524, 351988383031210413076295, 1474320303050934448138586, 6167313972271997160616487, 25770018446823167711177256, 107575128852482376189099657, 448686576996876913475900985, 1870064613139417627428681546, 7789228056784680467763728356, 32425967246106296890368810943, 134922331498272588364476180150, 561170234171421424687450762218, 2333185213162875626980569334586, 9697691681023767935816546925810, 40296761019115897693378020750304, 167405678599573178754554735425500, 695315826495982432201817860350384, 2887471697263577884599209836720724, 11989119731801937435908186367502418, 49773672878387017240820277186133933, 206615368239595050328432096365772786, 857596063782668973911429246019645248, 3559311146445642266628947699835442405, 14771247245703845390492597474797181501, 61297218039066894581942073485999795498, 254355134654745436101804689307395799176, 1055406241452059982356995468881303135245, 4379061349078358899285795579448995148357, 18168834136106060681393826933553149199771, 75380646388163385087709907289615387511431, 312738422596514964765543905180978445030357}.

L. E. Jeffery, Unit-primitive matrices

Cf. A205497, A050446, A050447.

G.f.: F(x) = (1 + 12*x - 112*x^2 - 343*x^3 + 3560*x^4 + 765*x^5 - 40847*x^6 + 10585*x^7 + 310877*x^8 - 193248*x^9 - 1419395*x^10 + 785781*x^11 + 5312667*x^12 - 2323912*x^13 - 15628824*x^14 + 5966469*x^15 + 33782788*x^16 - 10059915*x^17 - 55526776*x^18 + 8186536*x^19 + 73510769*x^20 + 2472617*x^21 - 80001340*x^22 - 15202136*x^23 + 70051834*x^24 + 21752017*x^25 - 47710282*x^26 - 20490103*x^27 + 24620158*x^28 + 14731526*x^29 - 9477868*x^30 - 8317984*x^31 + 2706852*x^32 + 3624852*x^33 - 575397*x^34 - 1176133*x^35 + 88180*x^36 + 269838*x^37 - 5571*x^38 - 39836*x^39 - 2463*x^40 + 2831*x^41 + 1104*x^42 + 107*x^43 - 221*x^44 - 36*x^45 + 23*x^46 + 2*x^47 - x^48) / ((1-x)^6 * (1-x-x^2)^5 * (1-2*x-x^2+x^3)^4 * (1-2*x-3*x^2+x^3+x^4)^3 * (1-3*x-3*x^2+4*x^3+x^4-x^5)^2 * (1-3*x-6*x^2+4*x^3+5*x^4-x^5-x^6)).
CONJECTURE 1. a(n) = M_{n,5} = M_{5,n}, where M = A205497.
CONJECTURE 2. Let w=2*cos(Pi/13). Then lim_{n->oo} a(n+1)/a(n) = w^5-4*w^3+3*w = spectral radius of the 6 X 6 unit-primitive matrix (see [Jeffery]) A_{13,5} = [0,0,0,0,0,1; 0,0,0,0,1,1; 0,0,0,1,1,1; 0,0,1,1,1,1; 0,1,1,1,1,1; 1,1,1,1,1,1].

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