Ran Pan - cp's OEIS frontend

A274764 Number of linear extensions of the one-level grid poset G[(1^n), (1^(n-1)), (1^(n-1))].

1, 20, 962, 75080, 8133732, 1127589120, 190416834360, 37902843124640, 8686847271179984, 2252403871470920960, 651771144516905730048, 208193858907016903262208, 72758882836839703611703296, 27613191886304138293279719424, 11308972154842887758316960743424, 4971172331379604809443266242019328

Offset: 1

Ran Pan, Jul 05 2016

nonn

The definition of a one-level grid poset can be found in the Pan links. The number of linear extensions of the one-level grid poset G[(0^n), (0^(n-1)), (0^(n-1))] is given by Catalan number A000108(n), the number of linear extensions of the one-level grid poset G[(1^n), (0^(n-1)), (0^(n-1))] is given by A274644(n) and the number of linear extensions of the one-level grid poset G[(1^n), (1^(n-1)), (0^(n-1))] is given by A274763(n).

Ran Pan, Problem 1, Project P.
Ran Pan, Algorithmic Solution to Problem 1 (and linear extensions of general one-level grid-like posets), Project P.

Cf. A000108, A274644, A274763.

A274763 Number of linear extensions of the one-level grid poset G[(1^n), (1^(n-1)), (0^(n-1))].

Original entry on oeis.org

1, 10, 215, 7200, 328090, 18914190, 1318595475, 107813147200, 10112867995550, 1070215246700100, 126122386636230950, 16378717184245443000, 2323753119238888045500, 357594668486650175355750, 59323244552378848484536875, 10553747415214416889115286000, 2004246729406751177924041663750, 404685181230584369889138573637500, 86569650968075614116679243211951250, 19558042902565983702641321883519060000

Offset: 1

Ran Pan, Jul 05 2016

nonn

The definition of a one-level grid poset can be found in the Pan links. The number of linear extensions of the one-level grid poset G[(0^n), (0^(n-1)), (0^(n-1))] is given by Catalan number A000108(n) and the number of linear extensions of the one-level grid poset G[(1^n), (0^(n-1)), (0^(n-1))] is given by A274644(n).

Cyril Banderier and Michael Wallner, Young Tableaux with Periodic Walls: Counting with the Density Method, Séminaire Lotharingien de Combinatoire, 85B (2021), Art. 47, 12 pp.
Ran Pan, Problem 1, Project P.
Ran Pan, Algorithmic Solution to Problem 1 (and linear extensions of general one-level grid-like posets), Project P.

Cf. A000108, A274644.

N := 100;
ff[1] := y-x;
for n from 1 to N-1 do
   ff[n+1] := simplify((y-x)*int(int((x-u)*subs(x=u,y=v,ff[n]),v=u..y),u=0..x));
end:
for n from 1 to N do
   a[n] := factorial(4*n-1)*int(int(ff[n],x=0..y),y=0..1);
end:
seq(a[n],n=1..10);
# Michael Wallner, Feb 14 2024

a(n) = (4*n-1)!*Integral_{y=0..1} Integral_{x=0..y} f_{n}(x,y) dx dy where f_{n+1}(x,y) = (y-x)*Integral_{u=0..x} Integral_{v=u..y} (x-u)*f_{n}(u,v) dv du for n>=1 and f_{1}(x,y) = y-x (Derived using the density method; see [Banderier, Wallner 2021]). - Michael Wallner, Feb 14 2024

Corrected and extended by Michael Wallner, Feb 14 2024

A274646 Number of linear extensions of the one-level grid poset G[(3^n), (0^(n-1)), (0^(n-1))].

Original entry on oeis.org

1, 70, 26599, 29609650, 72574079902, 332014782982540, 2545213373338499072, 30302687687176712355840, 529556871638491591748878336, 13004213964445490176628310933504, 433440210434110194677894532074307584

Offset: 1

Ran Pan, Jun 30 2016

nonn

The definition of a one-level grid poset can be found in the Pan links. The number of linear extensions of one-level grid poset G[(0^n), (0^(n-1)), (0^(n-1))] is given by Catalan number A000108(n), the number of linear extensions of the one-level grid poset G[(1^n), (0^(n-1)), (0^(n-1))] is given by A274644(n) and number of linear extensions of the one-level grid poset G[(2^n), (0^(n-1)), (0^(n-1))] is given by A274645(n).

Ran Pan, Problem 1, Project P.
Ran Pan, Algorithmic Solution to Problem 1 (and linear extensions of general one-level grid-like posets), Project P.

Cf. A000108, A274644, A274645.

A274645 Number of linear extensions of the one-level grid poset G[(2^n), (0^(n-1)), (0^(n-1))].

Original entry on oeis.org

1, 20, 1301, 177260, 41385102, 14760468600, 7465847167005, 5083351577582300, 4483012419041095680, 4971032496120058085376, 6769339545226095791964160, 11105730970797793499164966912, 21604722570792867452576610648064

Offset: 1

Ran Pan, Jun 30 2016

nonn

The definition of a one-level grid poset can be found in the Pan links. The number of linear extensions of the one-level grid poset G[(0^n), (0^(n-1)), (0^(n-1))] is given by the Catalan number A000108(n) and the number of linear extensions of the one-level grid poset G[(1^n), (0^(n-1)), (0^(n-1))] is given by A274644(n).

Ran Pan, Problem 1, Project P.
Ran Pan, Algorithmic Solution to Problem 1 (and linear extensions of general one-level grid-like posets), Project P.

Cf. A000108, A274644.

A274644 Number of linear extensions of the one-level grid poset G[(1^n), (0^(n-1)), (0^(n-1))].

Original entry on oeis.org

1, 6, 71, 1266, 30206, 902796, 32420011, 1359292626, 65164480466, 3515569641156, 210779736073446, 13903319821066836, 1000559812125494076, 78012524487061315416, 6550837823204594551731, 589404446176366002280146, 56568586570039148217467786, 5768723174387469795772704276, 622900652040379217092492454866

Offset: 1

Ran Pan, Jun 30 2016

nonn

The definition of a one-level grid poset can be found in the Pan links. The number of linear extensions of the one-level grid poset G[(0^n), (0^(n-1)), (0^(n-1))] is given by Catalan number A000108(n).

Michael Wallner, Table of n, a(n) for n = 1..100
Cyril Banderier and Michael Wallner, Young Tableaux with Periodic Walls: Counting with the Density Method, Séminaire Lotharingien de Combinatoire, 85B (2021), Art. 47, 12 pp.
Ran Pan, Problem 1, Project P.
Ran Pan, Algorithmic Solution to Problem 1 (and linear extensions of general one-level grid-like posets), Project P.

Cf. A000108, A274763.

M := 20;
for k from 3 to 3+2*M do
   bb[1,k] := 1;
end:
for n from 2 to M do
for k from 3 to 3+2*M-2*(n-1) do
   bb[n,k] := sum(i*bb[n-1,i+2],i=1..k);
end;
end:
seq(bb[n,3],n=1..10);
N := 100:
f[1] := y-x;
for n from 1 to N-1 do
   f[n+1] := (y-x)*int(int(subs(x=v,y=w,f[n]),w=v..y),v=0..x);
end:
for n from 1 to N do
   aa[n] := factorial(3*n)*int(int(f[n],x=0..y),y=0..1);
end:
seq(aa[n],n=1..10);
# Michael Wallner, Feb 13 2024

From Michael Wallner, Feb 13 2024: (Start)
a(n) = b(n,3) in b(n,k) = Sum_{i=1..k} i*b(n-1,i+2) for n>0 and k>=3 with initial conditions b(1,k) = 1 for all k.
a(n) = (3*n)!*Integral_{y=0..1} Integral_{x=0..y} f_{n}(x,y) dx dy where f_{n+1}(x,y) = (y-x)*Integral_{v=0..x} Integral_{w=v..y} f_{n}(v,w) dw dv for n>=1 and f_{1}(x,y) = y-x (Derived using the density method; see [Banderier, Wallner 2021]). (End)

All terms starting with a(13) corrected by Michael Wallner, Feb 13 2024

A268601 Expansion of 1/(2f(x)) - 1/(4 - 2g(x)), where f(x) = sqrt(1 - 4x) and g(x) = sqrt(1 + 4x).

Original entry on oeis.org

0, 0, 2, 8, 34, 120, 468, 1680, 6530, 23960, 93532, 348656, 1366260, 5149872, 20238696, 76907808, 302903874, 1158168792, 4569270156, 17555689008, 69356428284, 267518448912, 1058057586456, 4094231982048, 16208177203764, 62887835652720, 249156625186328, 968943740083040, 3841488520364200, 14968574892499040, 59379627044952528

Offset: 0

Ran Pan, Feb 08 2016

nonn

a(n) is the number of North-East lattice paths from (0,0) to (n,n) in which the total number of east steps below y = x - 1 or above y = x + 1 is odd. Details can be found in Section 4.1 in Pan and Remmel's link.

Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.

Cf. A268462, A268586, A268587, A268598, A268599.

x = 'x + O('x^30); concat(vector(2), Vec(1/(2*sqrt(1-4*x)) - 1/(4 - 2*sqrt(1+4*x)))) \\ Michel Marcus, Feb 11 2016

a(n) = binomial(2*n,n) - A268600(n).
G.f.: 1/(2*f(x)) - 1/(4 - 2*g(x)), where f(x) = sqrt(1 - 4*x) and g(x) = sqrt(1 + 4*x).
Conjecture D-finite with recurrence: 3*n*(n-1)*a(n) -8*(n-1)*(5*n-12)*a(n-1) +4*(28*n-73)*a(n-2) +160*(2*n-5)*(2*n-7)*a(n-3) -192*(2*n-5)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Jan 25 2020

A268600 Expansion of 1/(2f(x)) + 1/(4 - 2g(x)), where f(x) = sqrt(1 - 4x) and g(x) = sqrt(1 + 4x).

Original entry on oeis.org

1, 2, 4, 12, 36, 132, 456, 1752, 6340, 24660, 91224, 356776, 1337896, 5250728, 19877904, 78209712, 298176516, 1175437428, 4505865144, 17789574792, 68490100536, 270739425528, 1046041377264, 4139198745552, 16039426479336, 63522770785032, 246761907761776, 977995685565072, 3807202080396240, 15098691607042000, 58884954519908896

Offset: 0

Ran Pan, Feb 08 2016

nonn

a(n) is the number of North-East lattice paths from (0,0) to (n,n) in which the total number of east steps below y = x - 1 or above y = x + 1 is even. Details can be found in Section 4.1 in Pan and Remmel's link.

			G.f. = 1 + 2*x + 4*x^2 + 12*x^3 + 36*x^4 + 132*x^5 + 456*x^6 + ... - _Michael Somos_, May 16 2022

Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.

Cf. A268462, A268586, A268587, A268598, A268599.

Cf. A000984, A126984.

CoefficientList[ Series[1/(2 Sqrt[1 - 4x]) + 1/(4 - 2 Sqrt[1 + 4x]), {x, 0, 25}], x] (* Robert G. Wilson v, Nov 24 2016 *)

my(x = 'x + O('x^40)); Vec(1/(2*sqrt(1-4*x)) + 1/(4 - 2*sqrt(1+4*x))) \\ Michel Marcus, Feb 11 2016

a(n) = binomial(2*n,n) - A268601(n).
G.f.: 1/(2*f(x)) + 1/(4 - 2*g(x)), where f(x) = sqrt(1 - 4*x) and g(x) = sqrt(1 + 4*x).
Conjecture D-finite with recurrence: -3*n*(n-1)*a(n) +8*(n-1)*(5*n-12)*a(n-1) +4*(-28*n+73)*a(n-2) -160*(2*n-5)*(2*n-7)*a(n-3) +192*(2*n-5)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Jan 25 2020
a(n) = (-1)^n*A126984(n) + A268601(n). - Michael Somos, May 16 2022

A268598 Expansion of x^5(4 - 5x)/(1 - 2*x)^4.

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 27, 120, 440, 1440, 4368, 12544, 34560, 92160, 239360, 608256, 1517568, 3727360, 9031680, 21626880, 51249152, 120324096, 280166400, 647495680, 1486356480, 3391094784, 7693402112, 17364418560, 39007027200, 87241523200, 194330492928

Offset: 0

Ran Pan, Feb 08 2016

a(n) is the number of North-East lattice paths from (0,0) to (n,n) that have exactly two east steps below y = x - 1 and exactly one east step above y = x+1. Details can be found in Section 4.1 in Pan and Remmel's link.

Colin Barker, Table of n, a(n) for n = 0..1000
Ran Pan, Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Cf. A268462, A268586, A268587.

concat(vector(5), Vec((4-5*x)*x^5/(1-2*x)^4 + O(x^40))) \\ Michel Marcus, Feb 08 2016

G.f.: x^5*(4 - 5*x)/(1 - 2*x)^4.
From Colin Barker, Feb 08 2016: (Start)
a(n) = 2^(n-7)*(n-4)*(n-3)*(n+3) for n>2.
a(n) = 8*a(n-1)-24*a(n-2)+32*a(n-3)-16*a(n-4) for n>3. (End)

A268599 Expansion of 2x^6(4-10x+7x^2)/(1-2*x)^5.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 8, 60, 294, 1180, 4200, 13776, 42560, 125568, 357120, 985600, 2652672, 6988800, 18077696, 46018560, 115507200, 286326784, 701890560, 1703411712, 4096655360, 9771417600, 23132110848, 54384394240, 127049662464, 295069286400, 681574400000

Offset: 0

Ran Pan, Feb 08 2016

a(n) is the number of North-East lattice paths from (0,0) to (n,n) that have exactly two east steps below y = x - 1 and exactly two easts step above y = x + 1. Details can be found in Section 4.1 in Pan and Remmel's link.

Colin Barker, Table of n, a(n) for n = 0..1000
Ran Pan, Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (10,-40,80,-80,32).

Cf. A268462, A268586, A268587, A268598.

CoefficientList[Series[2 x^6 (4 - 10 x + 7 x^2)/(1 - 2 x)^5, {x, 0, 30}], x] (* Michael De Vlieger, Feb 08 2016 *)

concat(vector(6), Vec(2*x^6*(4-10*x+7*x^2)/(1-2*x)^5 + O(x^100))) \\ Colin Barker, Feb 08 2016

G.f.: 2*x^6*(4-10*x+7*x^2)/(1-2*x)^5.

a(n) = 2^(n-10)*(n-5)*(n-4)*(n^2+3*n+10) for n>3. - Colin Barker, Feb 08 2016

A268587 Expansion of x^4(5 - 16x + 13x^2)/(1 - 2x)^4.

Original entry on oeis.org

0, 0, 0, 0, 5, 24, 85, 264, 760, 2080, 5488, 14080, 35328, 87040, 211200, 505856, 1198080, 2809856, 6533120, 15073280, 34537472, 78643200, 178061312, 401080320, 899153920, 2006974464, 4461690880, 9881780224, 21810380800, 47982837760, 105243475968

Offset: 0

Ran Pan, Feb 07 2016

a(n) is the number of North-East lattice paths from (0,0) to (n,n) that have exactly three east steps below y = x - 1 and no east steps above y = x+1. Details can be found in Section 4.1 in Pan and Remmel's link.

Robert Israel, Table of n, a(n) for n = 0..3270
Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Cf. A268462, A268586.

Concatenation([0,0,0,0], List([3..40], n-> 2^(n-7)*(n-3)*(n+4)*(n+11)/3 )); # G. C. Greubel, May 24 2019

R:=PowerSeriesRing(Integers(), 40); [0,0,0,0] cat Coefficients(R!( x^4*(5-16*x+13*x^2)/(1-2*x)^4 )); // G. C. Greubel, May 24 2019

F:= gfun:-rectoproc({16*a(n)-32*a(n+1)+24*a(n+2)-8*a(n+3)+a(n+4), a(0)=0, a(1)=0,a(2)=0,a(3)=0,a(4)=5,a(5)=24,a(6)=85},a(n),remember):
map(F, [$0..40]); # Robert Israel, Feb 07 2016

CoefficientList[Series[x^4 (5 -16x +13x^2)/(1-2x)^4, {x, 0, 40}], x] (* Michael De Vlieger, Feb 08 2016 *)
LinearRecurrence[{8,-24,32,-16},{0,0,0,0,5,24,85},40] (* Harvey P. Dale, Feb 22 2025 *)

concat(vector(4), Vec(x^4*(5-16*x+13*x^2)/(1-2*x)^4 + O(x^40))) \\ Colin Barker, Feb 08 2016

(x^4*(5-16*x+13*x^2)/(1-2*x)^4).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 24 2019

G.f.: x^4*(5 - 16*x + 13*x^2)/(1 - 2*x)^4.
From Colin Barker, Feb 08 2016: (Start)
a(n) = 8*a(n-1) - 24*a(n-2) + 32*a(n-3) - 16*a(n-4) for n > 6.
a(n) = 2^(n-7)*(n-3)*(n+4)*(n+11)/3 for n > 2. (End)
E.g.f.: (33 + 60*x + 39*x^2 + (-33 + 6*x + 15*x^2 + 2*x^3)*exp(2*x))/96. - G. C. Greubel, May 24 2019

Typo in name and g.f. corrected by Georg Fischer, May 24 2019

cp's OEIS Frontend

User: Ran Pan

A274764 Number of linear extensions of the one-level grid poset G[(1^n), (1^(n-1)), (1^(n-1))].

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A274763 Number of linear extensions of the one-level grid poset G[(1^n), (1^(n-1)), (0^(n-1))].

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A274646 Number of linear extensions of the one-level grid poset G[(3^n), (0^(n-1)), (0^(n-1))].

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A274645 Number of linear extensions of the one-level grid poset G[(2^n), (0^(n-1)), (0^(n-1))].

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A274644 Number of linear extensions of the one-level grid poset G[(1^n), (0^(n-1)), (0^(n-1))].

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Maple

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A268601 Expansion of 1/(2*f(x)) - 1/(4 - 2*g(x)), where f(x) = sqrt(1 - 4*x) and g(x) = sqrt(1 + 4*x).

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A268600 Expansion of 1/(2*f(x)) + 1/(4 - 2*g(x)), where f(x) = sqrt(1 - 4*x) and g(x) = sqrt(1 + 4*x).

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A268598 Expansion of x^5*(4 - 5*x)/(1 - 2*x)^4.

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A268599 Expansion of 2*x^6*(4-10*x+7*x^2)/(1-2*x)^5.

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A268601 Expansion of 1/(2f(x)) - 1/(4 - 2g(x)), where f(x) = sqrt(1 - 4x) and g(x) = sqrt(1 + 4x).

A268600 Expansion of 1/(2f(x)) + 1/(4 - 2g(x)), where f(x) = sqrt(1 - 4x) and g(x) = sqrt(1 + 4x).

A268598 Expansion of x^5(4 - 5x)/(1 - 2*x)^4.

A268599 Expansion of 2x^6(4-10x+7x^2)/(1-2*x)^5.

A268587 Expansion of x^4(5 - 16x + 13x^2)/(1 - 2x)^4.