A277358 -id:A277358 - cp's OEIS frontend

A011781 Sextuple factorial numbers: a(n) = Product_{k=0..n-1} (6*k+3).

1, 3, 27, 405, 8505, 229635, 7577955, 295540245, 13299311025, 678264862275, 38661097149675, 2435649120429525, 168059789309637225, 12604484198222791875, 1020963220056046141875, 88823800144876014343125, 8260613413473469333910625, 817800727933873464057151875

Offset: 0

Lee D. Killough (killough(AT)wagner.convex.com)

Total number of Eulerian circuits in rooted labeled multigraphs with n edges. - Valery A. Liskovets, Apr 07 2002

Number of self-avoiding planar walks starting at (0,0), ending at (n,0), remaining in the east quadrant {(x,y): x >= |y|} and using steps (0,1), (0,-1), (1,1), (-1,-1), and (1,0). - Alois P. Heinz, Oct 13 2016

			G.f. = 1 + 3*x + 27*x^2 + 405*x^3 + 8505*x^4 + 229635*x^5 + 7577955*x^6 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Fatemeh Bagherzadeh, M. Bremner, and S. Madariaga, Jordan Trialgebras and Post-Jordan Algebras, arXiv:1611.01214 [math.RA], 2016.
Murray Bremner and Martin Markl, Distributive laws between the Three Graces, arXiv:1809.08191 [math.AT], 2018.
Bodo Lass, Démonstration combinatoire de la formule de Harer-Zagier, (A combinatorial proof of the Harer-Zagier formula) C. R. Acad. Sci. Paris, Serie I, 333(3) (2001), 155-160.
Bodo Lass, Démonstration combinatoire de la formule de Harer-Zagier, (A combinatorial proof of the Harer-Zagier formula) C. R. Acad. Sci. Paris, Serie I, Vol. 333, No. 3 (2001), pp. 155-160; alternative link.
Valery Liskovets, A Note on the Total Number of Double Eulerian Circuits in Multigraphs , Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.5.

Cf. A001147, A035309, A047657, A048994, A049308, A069736, A273464, A277358.

F:=Factorial;; List([0..20], n-> (3/2)^n*(F(2*n)/F(n)) ); # G. C. Greubel, Aug 20 2019

[(3/2)^n*Factorial(2*n)/Factorial(n):n in [0..20]]; // Vincenzo Librandi, May 09 2012

Table[Product[6k+3,{k,0,n-1}],{n,0,20}] (* or *) Table[6^(n-1) Pochhammer[ 1/2,n-1],{n,21}] (* Harvey P. Dale, May 09 2012 *)
Table[6^n*Pochhammer[1/2, n], {n,0,20}] (* G. C. Greubel, Aug 20 2019 *)

{a(n) = if( n<0, (-1)^n / a(-n), (3/2)^n * (2*n)! / n!)}; /* Michael Somos, Feb 10 2002, revised and extended Michael Somos, Jan 06 2017 */

[6^n*rising_factorial(1/2, n) for n in (0..20)] # G. C. Greubel, Aug 20 2019

E.g.f.: (1-6*x)^(-1/2).
a(n) = 3^n*(2*n-1)!!.
G.f.: 1/(1-3*x/(1-6*x/(1-9*x/(1-12*x/(1-15*x/(1-18*x/(1-21*x/(1-24*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-3)^n*Sum_{k=0..n} 2^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. [Mircea Merca, May 03 2012]
G.f.: T(0), where T(k) = 1 - 3*x*(k+1)/( 3*x*(k+1) - 1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 24 2013
a(n) = 6^n * gamma(n + 1/2) / sqrt(Pi). - Daniel Suteu, Jan 06 2017
0 = a(n)*(+6*a(n+1) - a(n+2)) + a(n+1)*(+a(n+1)) and a(n) = (-1)^n / a(-n) for all n in Z. - Michael Somos, Jan 06 2017
D-finite with recurrence: a(n) +3*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Jan 20 2018
From Amiram Eldar, Feb 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 1 + exp(1/6)*sqrt(Pi/6)*erf(1/sqrt(6)), where erf is the error function.
Sum_{n>=0} (-1)^n/a(n) = 1 - exp(-1/6)*sqrt(Pi/6)*erfi(1/sqrt(6)), where erfi is the imaginary error function. (End)

A284230 Number of self-avoiding planar walks starting at (0,0), ending at (n,0), remaining in the first quadrant and using steps (0,1), (1,0), (1,1), (-1,1), and (1,-1) with the restriction that (0,1) is never used below the diagonal and (1,0) is never used above the diagonal.

Original entry on oeis.org

1, 2, 5, 24, 111, 762, 5127, 45588, 400593, 4370634, 47311677, 611446464, 7857786015, 117346361778, 1745000283087, 29562853594284, 499180661754849, 9458257569095826, 178734707493557301, 3744942786114870888, 78294815164675006479, 1797384789345147560298

Offset: 0

Alois P. Heinz, Mar 23 2017

			a(0) = 1: [(0,0)].
a(1) = 2: [(0,0),(1,0)], [(0,0),(0,1),(1,0)].
a(2) = 5: [(0,0),(1,0),(2,0)], [(0,0),(0,1),(1,0),(2,0)], [(0,0),(1,1),(2,0)], [(0,0),(0,1),(0,2),(1,1),(2,0)], [(0,0),(1,0),(0,1),(0,2),(1,1),(2,0)].

Alois P. Heinz, Table of n, a(n) for n = 0..448
Alois P. Heinz, Animation of a(4)=111 walks
Wikipedia, Lattice path
Wikipedia, Self-avoiding walk

Row sums of A284414.
Bisection (even part) gives A284461.
Cf. A001900, A277358, A284231, A285673.

a:= proc(n) option remember; `if`(n<2, n+1,
      (n+irem(n, 2))*a(n-1)+(n-1)*a(n-2))
    end:
seq(a(n), n=0..25);

a[n_]:=If[n<2, n + 1, (n + Mod[n,2]) * a[n - 1] + (n - 1) a[n - 2]]; Table[a[n], {n, 0, 25}] (* Indranil Ghosh, Mar 27 2017 *)

a(n) ~ c * n^(n+2) / exp(n), where c = 0.7741273379869056907732932906458364317717498069987762339667734187318... - Vaclav Kotesovec, Mar 27 2017

Conjecture: a(n) -a(n-1) +(-n^2-n+3)*a(n-2) +(-n+2)*a(n-3) +(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Apr 09 2017

A277359 Number of self-avoiding planar walks starting at (0,0), ending at (n,n), remaining in the first quadrant and using steps (0,1) and (1,0) on or below the diagonal and using steps (1,1), (-1,1), and (1,-1) on or above the diagonal.

Original entry on oeis.org

1, 2, 7, 32, 176, 1126, 8227, 67768, 622706, 6323932, 70400734, 852952952, 11176241098, 157506733030, 2375966883371, 38200984291800, 652179787654530, 11783182484950980, 224623760504277810, 4505795627243046240, 94873821120923655336, 2092249161797280567516

Offset: 0

Alois P. Heinz, Oct 10 2016

Both endpoints of each step have to satisfy the given restrictions.

a(n) is odd for n in {0, 2, 6, 14, 30, 62, ... } = { 2^n-2 | n>0 }.

Alois P. Heinz, Table of n, a(n) for n = 0..448

Cf. A000108, A000142, A000918, A277175, A277176, A277358, A277360, A277756.

a:= proc(n) option remember; `if`(n<3, [1, 2, 7][n+1],
      ((n^3+10*n^2-10*n+1)*a(n-1)-(2*(4*n^3+2*n^2-29*n+28))
        *a(n-2)+(4*(n-2))*(2*n-3)^2*a(n-3))/(n*(n+1)))
    end:
seq(a(n), n=0..25);

a[n_] := a[n] = If[n<3, {1, 2, 7}[[n+1]], ((n^3+10*n^2-10*n+1)*a[n-1] - (2*(4*n^3+2*n^2-29*n+28))*a[n-2] + (4*(n-2))*(2*n-3)^2*a[n-3])/(n*(n+1)) ]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 25 2017, translated from Maple *)

a(n) ~ exp(1)*(exp(1)-2) * n! * n. - Vaclav Kotesovec, Oct 13 2016

A277424 Total number of nodes summed over all self-avoiding planar walks starting at (0,0), ending at (n,0), remaining in the first quadrant and using steps (0,1), (1,0), (1,1), (-1,1), and (1,-1).

Original entry on oeis.org

1, 5, 39, 379, 4457, 61503, 974107, 17412317, 346662981, 7605810685, 182298744203, 4738700778123, 132767583248917, 3988244997744743, 127859570155253607, 4357113615504651565, 157266354405499307369, 5993377455733610208885, 240479249123008267155343

Offset: 0

Alois P. Heinz, Oct 14 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A277358, A284231, A285673.

b:= proc(x, y, t) option remember; `if`(x<0 or y<0, 0,
      `if`(x=0 and y=0, [1$2], (p-> p+ [0, p[1]])(
       b(x-1, y, 0)+ b(x, y-1, 0)+ b(x-1, y-1, 0)+
      `if`(t<>2, b(x+1, y-1, 1), 0)+
      `if`(t<>1, b(x-1, y+1, 2), 0))))
    end:
a:= n-> b(n, 0$2)[2]:
seq(a(n), n=0..25);

b[x_, y_, t_] := b[x, y, t] = If[x < 0 || y < 0, 0, If[x == 0 && y == 0, {1, 1}, Function[p, p + {0, p[[1]]}][b[x - 1, y, 0] + b[x, y - 1, 0] + b[x - 1, y - 1, 0] + If[t != 2, b[x + 1, y - 1, 1], 0] + If[t != 1, b[x - 1, y + 1, 2], 0]]]];
a[n_] := b[n, 0, 0][[2]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 19 2017, translated from Maple *)

From Vaclav Kotesovec, Oct 14 2016: (Start)
Recurrence: (n^3 + n^2 - 6*n + 1)*a(n) = (4*n^4 + 6*n^3 - 28*n^2 + 3*n + 4)*a(n-1) - (4*n^5 + 2*n^4 - 42*n^3 + 50*n^2 - 13)*a(n-2) - (n-2)*(4*n^4 + 8*n^3 - 34*n^2 - n + 16)*a(n-3) - (n-3)*(n-2)*(n^3 + 4*n^2 - n - 3)*a(n-4).
a(n) ~ sqrt(Pi) * 2^(n+3/2) * n^(n+11/4) / (3 * Gamma(1/4) * exp(n+1/4)).
(End)

A277360 Number of self-avoiding planar walks starting at (0,0), ending at (n,n), remaining in the first quadrant and using steps (0,1), (1,0), (1,1), (-1,1), and (1,-1).

Original entry on oeis.org

1, 9, 491, 64159, 15314361, 5799651689, 3193954129651, 2410542221526399, 2388182999073694001, 3006071549433968619529, 4685653563347872021885371, 8859314350383162594502273439, 19975392290718104323103596377961, 52949467092712165429316121638458089

Offset: 0

Alois P. Heinz, Oct 10 2016

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A277358, A277359.

a:= proc(n) option remember; `if`(n<2, 8*n+1,
      (16*n^2-4*n-1)*a(n-1)-n*(4*n-6)*a(n-2))
    end:
seq(a(n), n=0..15);

a[n_] := a[n] = If[n<2, 8n+1, (16n^2 - 4n - 1) a[n-1] - n (4n-6) a[n-2]];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Mar 29 2017, translated from Maple *)

a(n) = (16*n^2-4*n-1)*a(n-1) - n*(4*n-6)*a(n-2) for n>1, a(0)=1, a(1)=9.
a(n) = (2n)! * [x^(2n)] exp(-x/2)/(1-2*x)^(5/4).
a(n) = A277358(2*n).
a(n) ~ sqrt(Pi) * 2^(4*n + 13/4) * n^(2*n + 3/4) / (Gamma(1/4) * exp(2*n + 1/4)). - Vaclav Kotesovec, Oct 13 2016

A317985 Number of self-avoiding planar walks starting at (0,0), ending at (n,0), remaining in the first quadrant and using steps (0,1), (1,0), (1,1), (-1,1), and (1,-1) such that (0,1) is never used directly before or after (1,0) or (1,1).

Original entry on oeis.org

1, 2, 7, 38, 284, 2691, 30890, 416449, 6448243, 112751661, 2197200541, 47214026822, 1109022356759, 28269085769331, 777140210643254, 22918982645377342, 721764216387297451, 24173661551378798838, 857993099925433301350, 32168967331652245055171

Offset: 0

Alois P. Heinz, Oct 02 2018

Alois P. Heinz, Table of n, a(n) for n = 0..403

Cf. A277358, A320512.

a:= proc(n) option remember; `if`(n<4, [1, 2, 7, 38][n+1],
      2*n*a(n-1) -(n-2)*a(n-2) -(2*n-5)*a(n-3))
    end:
seq(a(n), n=0..25);

a = DifferenceRoot[Function[{y, n}, {(2n+1) y[n] + (n+1) y[n+1] + (-2n-6)* y[n+2] + y[n+3] == 0, y[0] == 1, y[1] == 2, y[2] == 7, y[3] == 38}]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 12 2020, after Maple *)
nmax = 20; CoefficientList[Simplify[Normal[Series[-1 - 1/x^(3/4) * E^(-1/(2*x) + (3*ArcTanh[(1 + 4*x)/Sqrt[17]])/(4*Sqrt[17]))* (-2 + x + 2*x^2)^(1/8) * Integrate[E^(1/(2*x)) * Simplify[Normal[Series[(-2 + 2*x + x^2)/(x^(5/4)*(-2 + x + 2*x^2)^(9/8))/ E^(3*ArcTanh[(1 + 4*x)/Sqrt[17]] / (4*Sqrt[17])), {x, 0, nmax}]], x > 0], x], {x, 0, nmax}]], x > 0], x] (* Vaclav Kotesovec, May 14 2020 *)

a(n) ~ c * 2^n * n! / n^(1/4), where c = 1.054816768531988358301631965137203014379828345839423725829486842843413035459... - Vaclav Kotesovec, May 14 2020

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A011781 Sextuple factorial numbers: a(n) = Product_{k=0..n-1} (6*k+3).

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A284230 Number of self-avoiding planar walks starting at (0,0), ending at (n,0), remaining in the first quadrant and using steps (0,1), (1,0), (1,1), (-1,1), and (1,-1) with the restriction that (0,1) is never used below the diagonal and (1,0) is never used above the diagonal.

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A277359 Number of self-avoiding planar walks starting at (0,0), ending at (n,n), remaining in the first quadrant and using steps (0,1) and (1,0) on or below the diagonal and using steps (1,1), (-1,1), and (1,-1) on or above the diagonal.

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A277424 Total number of nodes summed over all self-avoiding planar walks starting at (0,0), ending at (n,0), remaining in the first quadrant and using steps (0,1), (1,0), (1,1), (-1,1), and (1,-1).

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A277360 Number of self-avoiding planar walks starting at (0,0), ending at (n,n), remaining in the first quadrant and using steps (0,1), (1,0), (1,1), (-1,1), and (1,-1).

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A317985 Number of self-avoiding planar walks starting at (0,0), ending at (n,0), remaining in the first quadrant and using steps (0,1), (1,0), (1,1), (-1,1), and (1,-1) such that (0,1) is never used directly before or after (1,0) or (1,1).

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