A086405 -id:A086405 - cp's OEIS frontend

A206948 Number of nonisomorphic graded posets with 0 and non-uniform Hasse graph of rank n, with exactly 2 elements of each rank level above 0.

0, 0, 0, 2, 19, 131, 791, 4446, 23913, 124892, 638878, 3218559, 16027375, 79093773, 387540260, 1887974063, 9154751912, 44221373872, 212931964415, 1022594028515, 4900116587043, 23437066655010, 111923110602497

Offset: 0

David Nacin, Feb 13 2012

We do not assume all maximal elements have maximal rank and thus use graded poset to mean: For every element x, all maximal chains among those with x as greatest element have the same finite length. Here, the term uniform used in the sense of Retakh, Serconek and Wilson.

R. Stanley, Enumerative combinatorics. Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
V. Retakh, S. Serconek, and R. Wilson, Hilbert Series of Algebras Associated to Directed Graphs and Order Homology, arXiv:1010.6295 [math.RA], 2010-2011.
Wikipedia, Graded poset
Index entries for linear recurrences with constant coefficients, signature (11, -40, 55, -30, 6).

a(n) = A086405(n) - A012781(n+1).
Cf. A206947 (unique maximal element added).
Cf. A206949, A206950 (allowing one or two elements in each rank level above 0 with and without maximal element).

LinearRecurrence[{11, -40, 55, -30, 6}, {0, 0, 0, 2, 19, 131}, 23] (* David Nacin, Feb 29 2012; a(0) added by Georg Fischer, Apr 03 2019 *)

def a(n, adict={0:0, 1:0, 2:0, 3:2, 4:19, 5:131}):
  if n in adict:
    return adict[n]
  adict[n]=11*a(n-1)-40*a(n-2)+55*a(n-3)-30*a(n-4)+6*a(n-5)
  return adict[n]
for n in range(0,40):
  print(a(n))

a(n) = 11*a(n-1) - 40*a(n-2) + 55*a(n-3) - 30*a(n-4) + 6*a(n-5), a(0)=0, a(1)=0, a(2)=0, a(3)=2, a(4)=19, a(5)=131.

G.f.: (x^3*(2 - 3*x + 2*x^2))/((1 - 6*x + 6*x^2)*(1 - 5*x + 4*x^2 - x^3)).

A110210 a(n+3) = 6a(n) - 5a(n+2), a(0) = -1, a(1) = 1, a(2) = -5.

Original entry on oeis.org

-1, 1, -5, 19, -89, 415, -1961, 9271, -43865, 207559, -982169, 4647655, -21992921, 104071591, -492472025, 2330402599, -11027583449, 52183085095, -246933009881, 1168499548711, -5529399232985, 26165398105639, -123815993235929, 585903570781735, -2772525465274841

Offset: 0

Creighton Dement, Jul 16 2005

Index entries for linear recurrences with constant coefficients, signature (-5,0,6).

Cf. A086405, A094433, A110211, A110212, A110213.

seriestolist(series((1+4*x)/((x-1)*(6*x^2+6*x+1)), x=0,25));

LinearRecurrence[{-5,0,6},{-1,1,-5},30] (* or *) CoefficientList[ Series[ (1+4*x)/(-1-5*x+6*x^3),{x,0,30}],x] (* Harvey P. Dale, Nov 09 2014 *)

Superseeker finds: a(n+1) - a(n) = ((-1)^n)*A094433(n+2); a(n+2) - a(n) = ((-1)^(n+1))*A086405(n+1).

G.f.: (4*x+1)/(6*x^3-5*x-1). - Harvey P. Dale, Nov 09 2014

A162557 a(n) = ((3+sqrt(3))(4+sqrt(3))^n+(3-sqrt(3))(4-sqrt(3))^n)/6.

Original entry on oeis.org

1, 5, 27, 151, 857, 4893, 28003, 160415, 919281, 5268853, 30200171, 173106279, 992248009, 5687602445, 32601595443, 186873931759, 1071170713313, 6140004593637, 35194817476027, 201738480090935, 1156375213539129, 6628401467130877, 37994333961038339, 217785452615605311

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 06 2009

nonn

Binomial transform of A086405.
Inverse binomial transform of A162558.
4th binomial transform of A108411.
2nd binomial transform of A079935. [R. J. Mathar, Jul 17 2009]
From J. Conrad, Aug 29 2016: (Start)
Partial sum of A136777.
Backward difference of Sum_{k=0..n} A027907(n+1,2k+2)*3^k.
(End)
String length in substitution system {0 -> 1001001, 1 -> 11011} at step n from initial string "1" (1 -> 11011 -> 110111101110010011101111011 -> ...). - Ilya Gutkovskiy, Aug 30 2016

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-13).

Cf. A108411 (powers of 3 repeated), A086405, A162558.

Cf. A162558. [R. J. Mathar, Jul 17 2009]

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((3+r)*(4+r)^n+(3-r)*(4-r)^n)/6: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 13 2009

I:=[1,5]; [n le 2 select I[n]  else 8*Self(n-1)-13*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Aug 30 2016

seq(simplify(((3+sqrt(3))*(4+sqrt(3))^n+(3-sqrt(3))*(4-sqrt(3))^n)*1/6), n = 0..20); # Emeric Deutsch, Jul 14 2009

Table[FullSimplify[((3 + #) (4 + #)^n + (3 - #) (4 - #)^n)/6 &@ Sqrt@ 3], {n, 0, 23}] (* Michael De Vlieger, Aug 30 2016 *)
LinearRecurrence[{8,-13},{1,5},30] (* Harvey P. Dale, Oct 23 2020 *)

a(n) = 8*a(n-1)-13*a(n-2) for n > 1; a(0) = 1, a(1) = 5.

G.f.: (1-3*x)/(1-8*x+13*x^2).

Edited, corrected and extended beyond a(5) by Klaus Brockhaus, Emeric Deutsch and R. J. Mathar, Jul 07 2009

More terms from Vincenzo Librandi, Aug 30 2016

A086404 Square array of numbers T(n,k) = ((1+sqrt(3))(k+sqrt(3))^n-(1-sqrt(3))(k-sqrt(3))^n)/(2*sqrt(3)), read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 3, 1, 4, 11, 16, 9, 1, 5, 18, 41, 44, 9, 1, 6, 27, 84, 153, 120, 27, 1, 7, 38, 151, 396, 571, 328, 27, 1, 8, 51, 248, 857, 1872, 2131, 896, 81, 1, 9, 66, 381, 1644, 4893, 8856, 7953, 2448, 81, 1, 10, 83, 556, 2889, 10984, 28003, 41904, 29681

Offset: 0

Paul Barry, Jul 19 2003

			Rows begin
  1, 1,  3,   3,   9, ...
  1, 2,  6,  16,  44, ...
  1, 3, 11,  41, 153, ...
  1, 4, 18,  84, 396, ...
  1, 5, 27, 151, 857, ...

Rows include A002605, A079935, A086405. Main diagonal is A086406. Rows are successive binomial transforms of (1, 1, 3, 3, 9, 9, ...).

Cf. A086350.

A162558 a(n) = ((3+sqrt(3))(5+sqrt(3))^n + (3-sqrt(3))(5-sqrt(3))^n)/6.

Original entry on oeis.org

1, 6, 38, 248, 1644, 10984, 73672, 495072, 3329936, 22407776, 150819168, 1015220608, 6834184384, 46006990464, 309717848192, 2085024691712, 14036454256896, 94493999351296, 636137999861248, 4282512012883968

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 06 2009

nonn

Fifth binomial transform of A108411. Binomial transform of A162557. Inverse binomial transform of A162757.

2nd binomial transform of A086405. - R. J. Mathar, Jul 17 2009

Index entries for linear recurrences with constant coefficients, signature (10, -22).

Cf. A108411 (powers of 3 repeated), A162557, A162757.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((3+r)*(5+r)^n+(3-r)*(5-r)^n)/6: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 13 2009

a(n) = 10*a(n-1) - 22*a(n-2) for n > 1; a(0) = 1, a(1) = 6.
G.f.: (1-4*x)/(1-10*x+22*x^2).
From R. J. Mathar, Jul 17 2009: (Start)
a(n) = 10*a(n-2) - 22*a(n-2).
G.f.: (1-4*x)/(1-10*x+22*x^2). (End)

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 13 2009

More terms from R. J. Mathar, Jul 17 2009

A208736 Number of nonisomorphic graded posets with 0 and 1 and non-uniform Hasse graph of rank n, with exactly 2 elements of each rank level between 0 and 1.

Original entry on oeis.org

0, 0, 0, 1, 5, 22, 91, 361, 1392, 5265, 19653, 72694, 267179, 977593, 3565600, 12975457, 47142021, 171075606, 620303547, 2247803785, 8141857808, 29481675889, 106728951109, 386314552438, 1398132674955, 5059626441177, 18308871648576, 66249898660801

Offset: 0

David Nacin, Mar 01 2012

Uniform used in the sense of Retakh, Serconek and Wilson. We use Stanley's definition of graded poset: all maximal chains have the same length n (which also implies all maximal elements have maximal rank.)

R. Stanley, Enumerative combinatorics. Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
V. Retakh, S. Serconek, and R. Wilson, Hilbert Series of Algebras Associated to Directed Graphs and Order Homology, arXiv:1010.6295 [math.RA], 2010-2011.
Wikipedia, Graded poset
Index entries for linear recurrences with constant coefficients, signature (8,-21,20,-5).

Cf. A208737, A206901, A206902, A206947-A206950, A001906, A025192, A081567, A124302, A124292, A088305, A086405, A012781.

Join[{0, 0}, LinearRecurrence[{8, -21, 20, -5}, {0, 1, 5, 22}, 40]]

def a(n, d={0:0,1:0,2:0,3:1,4:5,5:22}):
    if n in d:
        return d[n]
    d[n]=8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4)
    return d[n]

a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4), a(2) = 0, a(3) = 1, a(4) = 5, a(5) = 22.
G.f.: (x^3 - 3*x^4 + 3*x^5)/(1 - 8*x + 21*x^2 - 20*x^3 + 5*x^4); (x^3 * (1 - 3*x + 3*x^2))/((1 - 3*x + x^2)*(1 - 5*x + 5*x^2)) .
a(n) = A081567(n-2) - A001519(n-1).

A208737 Number of nonisomorphic graded posets with 0 and 1 and non-uniform Hasse graph of rank n, with no 3-element antichain.

Original entry on oeis.org

0, 0, 0, 1, 7, 37, 175, 778, 3325, 13837, 56524, 227866, 909832, 3607294, 14227447, 55894252, 218937532, 855650749, 3338323915, 13007422705, 50631143323, 196928737582, 765495534433, 2974251390529, 11552064922624, 44856304154086

Offset: 0

David Nacin, Mar 01 2012

Uniform used in the sense of Retakh, Serconek and Wilson. We use Stanley's definition of graded poset: all maximal chains have the same length n (which also implies all maximal elements have maximal rank.)

R. Stanley, Enumerative combinatorics. Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
V. Retakh, S. Serconek, and R. Wilson, Hilbert Series of Algebras Associated to Directed Graphs and Order Homology, arXiv:1010.6295 [math.RA], 2010-2011.
Wikipedia, Graded poset
Index entries for linear recurrences with constant coefficients, signature (10,-36,57,-39,9).

Cf. A208736, A206901, A206902, A206947-A206950, A001906, A025192, A081567, A124302, A124292, A088305, A086405, A012781.

Join[{0}, LinearRecurrence[{10, -36, 57, -39, 9}, {0, 0, 1, 7, 37}, 40]]

def a(n, d={0:0,1:0,2:0,3:1,4:7,5:37}):
    if n in d:
        return d[n]
    d[n]=10*a(n-1) - 36*a(n-2) + 57*a(n-3) - 39*a(n-4) + 9*a(n-5)
    return d[n]

a(n) = 10*a(n-1) - 36*a(n-2) + 57*a(n-3) - 39*a(n-4) + 9*a(n-5), a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 7, a(5) = 37.
G.f: (x^3 - 3*x^4 + 3*x^5)/(1 - 10*x + 36*x^2 - 57*x^3 + 39*x^4 - 9*x^5); (x^3*(1 - 3*x + 3*x^2)) / ((1 - x) (1 - 3*x) (1 - 6*x + 9*x^2 - 3*x^3)).
a(n) = A124292(n) - A124302(n).

cp's OEIS Frontend

A206948 Number of nonisomorphic graded posets with 0 and non-uniform Hasse graph of rank n, with exactly 2 elements of each rank level above 0.

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A110210 a(n+3) = 6*a(n) - 5*a(n+2), a(0) = -1, a(1) = 1, a(2) = -5.

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A162557 a(n) = ((3+sqrt(3))*(4+sqrt(3))^n+(3-sqrt(3))*(4-sqrt(3))^n)/6.

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A086404 Square array of numbers T(n,k) = ((1+sqrt(3))*(k+sqrt(3))^n-(1-sqrt(3))*(k-sqrt(3))^n)/(2*sqrt(3)), read by antidiagonals.

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A162558 a(n) = ((3+sqrt(3))*(5+sqrt(3))^n + (3-sqrt(3))*(5-sqrt(3))^n)/6.

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A208736 Number of nonisomorphic graded posets with 0 and 1 and non-uniform Hasse graph of rank n, with exactly 2 elements of each rank level between 0 and 1.

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A208737 Number of nonisomorphic graded posets with 0 and 1 and non-uniform Hasse graph of rank n, with no 3-element antichain.

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Formula

A110210 a(n+3) = 6a(n) - 5a(n+2), a(0) = -1, a(1) = 1, a(2) = -5.

A162557 a(n) = ((3+sqrt(3))(4+sqrt(3))^n+(3-sqrt(3))(4-sqrt(3))^n)/6.

A086404 Square array of numbers T(n,k) = ((1+sqrt(3))(k+sqrt(3))^n-(1-sqrt(3))(k-sqrt(3))^n)/(2*sqrt(3)), read by antidiagonals.

A162558 a(n) = ((3+sqrt(3))(5+sqrt(3))^n + (3-sqrt(3))(5-sqrt(3))^n)/6.