A284230 -id:A284230 - cp's OEIS frontend

A284414 Number T(n,k) of self-avoiding planar walks of length k 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; triangle T(n,k), n>=0, n<=k<=n*(n+3)/2, read by rows.

1, 1, 1, 2, 1, 1, 1, 4, 4, 4, 7, 3, 1, 1, 9, 8, 16, 21, 17, 15, 10, 9, 4, 1, 1, 21, 22, 54, 87, 87, 116, 99, 91, 78, 42, 31, 17, 10, 5, 1, 1, 51, 54, 178, 269, 370, 499, 536, 590, 560, 510, 420, 350, 268, 185, 132, 69, 44, 23, 11, 6, 1, 1

Offset: 0

Alois P. Heinz, Mar 26 2017

			Triangle T(n,k) begins:
1;
.  1, 1;
.  .  2, 1, 1,  1;
.  .  .  4, 4,  4,  7,  3,  1,  1;
.  .  .  .  9,  8, 16, 21, 17, 15,  10,  9,  4,  1,  1;
.  .  .  .  .  21, 22, 54, 87, 87, 116, 99, 91, 78, 42, 31, 17, 10, 5, 1, 1;

Alois P. Heinz, Rows n = 0..50, flattened
Alois P. Heinz, Animation of T(5,12)=91 walks
Wikipedia, Lattice_path
Wikipedia, Self-avoiding walk

Row sums give A284230.
Column sums give A284415.
Antidiagonal sums give A284428.
T(n,n) gives A001006.
T(n,n+1) gives A284778.
T(n,2n) gives A284416.
T(n,n*(n+1)/2) gives A284418.
Cf. A000096, A284231, A284461, A284652 (this triangle read by columns).

Sum_{k=n..n*(n+3)/2} (k+1) * T(n,k) = A284231(n).

A001900 Successive numerators of Wallis's approximation to Pi/2 (unreduced).

Original entry on oeis.org

1, 2, 4, 16, 64, 384, 2304, 18432, 147456, 1474560, 14745600, 176947200, 2123366400, 29727129600, 416179814400, 6658877030400, 106542032486400, 1917756584755200, 34519618525593600, 690392370511872000, 13807847410237440000, 303772643025223680000

Offset: 0

N. J. A. Sloane

a(n) = number of permutations of [n+1] all of whose non-initial left-to-right minima are at even positions in the permutation. For example, a(2) = 4 counts 123, 132, 213, 312. - David Callan, Jul 22 2008
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), and (1,-1) with the restriction that (0,1) is never used below the diagonal and (1,0) is never used above the diagonal. a(2) = 4: [(0,0),(1,0),(2,0)], [(0,0),(0,1),(1,0),(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, Mar 23 2017
a(n+1) is the number of 0-1 square matrices of order n+1 with 2n+1 nonzero entries where the cell (i,j) is 1 for all i+j=n+2 and every diagonal, parallel to the main diagonal, has exactly one 1. For example, a(2) = 4: [(0,1,1), (1,1,0), (1,0,0)], [(0,1,1), (0,1,0), (1,1,0)], [(0,0,1), (1,1,1), (1,0,0)], [(0,0,1), (0,1,1), (1,1,0)]. - Christian Barrientos, Jul 17 2021

			From _Wolfdieter Lang_, Dec 06 2017: (Start)
Partial products of the rows N (for numerators a(n)) and D (for denominators b(n) = A000246(n+1)) begin:
n:    0  1  2  3  4   5    6     7      8       9       10 ...
N:    1  2  2  4  4   6    6     8      8      10       10 ...
D:    1  1  3  3  5   5    7     7      9       9       11 ...
a(n): 1  2  4 16 64 384 2304 18432 147456 14745601 4745600 ...
b(n): 1  1  3  9 45 225 1575 11025  99225   893025 9823275 ...  (End)

H.-D. Ebbinghaus et al., Numbers, Springer, 1990, p. 146.

Alois P. Heinz, Table of n, a(n) for n = 0..449
John Derbyshire, Prime Obsession, Plume books, p. 16, 2003.
Jonathan Sondow, A faster product for Pi and a new integral for ln(Pi/2), arXiv:math/0401406 [math.NT], 2004.
Jonathan Sondow, A faster product for Pi and a new integral for ln(Pi/2), Amer. Math. Monthly 112 (2005), 729-734 and 113 (2006), 670.
Index to divisibility sequences

Cf. A000246, A284230.

For the reduced form see A001901(n)/A001902(n), n >= 0.

a[n_] := a[n] = If[n==0, 1, (n+Mod[n, 2]) a[n-1]];
a /@ Range[0, 21] (* Jean-François Alcover, Jan 31 2020 *)

a(n)=if(n<0,0,prod(k=1,n,if(k%2,k+1,k)))

2.2.4.4.6.6....2n.2n.../1.3.3.5.5.7.7....(2n-1).(2n+1) ...for n >= 1.
a(n) = 2^n * A010551(n) = 2^n * [n/2]! * [(n+1)/2]!. - Ralf Stephan, Mar 11 2004
Conjecture: a(n) - a(n-1) - n*(n-1)*a(n-2) = 0. - R. J. Mathar, Jun 07 2013 [The proof, for n >= 2, follows from the bisection recurrence given below. - Wolfdieter Lang, Dec 07 2017]
E.g.f.: E(0), where E(k)= 1 + 2*x*(k+1)/((2*k+1) - x*(2*k+1)/(x + 1/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 08 2013
G.f.: G(0), where G(k)= 1 + 2*x*(k+1)/(1 - 2*x*(k+1)/(2*x*(k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 08 2013
Bisection: a(2*k+1) = ((2*k+1)+1)*a(2*k), a(2*k) = 2*k*a(2*k-1), k >= 0, with a(0) = 1. The proof is obvious from the numbers in the numerator (see the row N in the example). From a proposal by David James Sycamore, Nov 02 2017 based on the fractions 4/1, 8/3, 32/9, 128/45, ... converging very slowly to Pi, given on p. 16 of the Derbyshire link. - Wolfdieter Lang, Dec 06 2017
From Mathias Zechmeister, Jul 24 2022: (Start)
Let J_0(x) and J_1(x) denote the Bessel functions and i = sqrt(-1).
a(n) = denominator([x^n](J_0(x) + J_1(x))).
a(n) = denominator([x^n](J_0(i*x) - i*J_1(i*x))).
Bisection: a(2*k) = A002454(k), a(2*k+1) = A002474(k).
G.f. for 1/a(n): J_0(i*x) - i*J_1(i*x).  (End)

A277358 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).

Original entry on oeis.org

1, 2, 9, 58, 491, 5142, 64159, 929078, 15314361, 283091122, 5799651689, 130423248378, 3193954129651, 84607886351462, 2410542221526399, 73500777054712438, 2388182999073694001, 82374234401380995042, 3006071549433968619529, 115713455097715665452858

Offset: 0

Alois P. Heinz, Oct 10 2016

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

Cf. A277359, A277360, A277424, A284230, A317985.

a:= n-> n!*coeff(series(exp(-x/2)/(1-2*x)^(5/4), x, n+1), x, n):
seq(a(n), n=0..25);
# second Maple program:
a:= proc(n) option remember; `if`(n<2, n+1,
       2*n*a(n-1) +(n-1)*a(n-2))
    end:
seq(a(n), n=0..25);

a[n_] := a[n] = If[n < 2, n+1, 2*n*a[n-1] + (n-1)*a[n-2]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 29 2017, translated from Maple *)

E.g.f.: exp(-x/2)/(1-2*x)^(5/4).
a(n) = 2*n*a(n-1) + (n-1)*a(n-2) for n>1, a(0)=1, a(1)=2.
a(n) ~ sqrt(Pi) * 2^(n+5/2) * n^(n+3/4) / (Gamma(1/4) * exp(n+1/4)). - Vaclav Kotesovec, Oct 13 2016

A284231 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) 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, 5, 21, 152, 975, 8835, 75499, 830180, 8819417, 114384573, 1450018173, 21689509992, 319180726887, 5411092531323, 90615453774771, 1717272516535812, 32234085990345105, 675335923050095253, 14040521125141683717, 322252846702242056280, 7349647183279936080543

Offset: 0

Alois P. Heinz, Mar 23 2017

			a(0) = 1: [(0,0)].
a(1) = 5: [(0,0),(1,0)], [(0,0),(0,1),(1,0)].
a(2) = 21: [(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..446
Wikipedia, Lattice path
Wikipedia, Self-avoiding walk

Cf. A277424, A284230, A284414, A285673.

a(n) = Sum_{k=n..n*(n+3)/2} (k+1) * A284414(n,k).

A285673 Total number of nodes summed over all 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) 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, 20, 907, 69928, 8190329, 1352590668, 299134112595, 85301875065360, 30466886170947633, 13319092946564641476, 6994728861780241970523, 4344874074153003071077560, 3150737511338249699332032297, 2637670112785000275509973725820, 2524664376417193478764383143006883

Offset: 0

Alois P. Heinz, Apr 24 2017

Alois P. Heinz, Table of n, a(n) for n = 0..223
Wikipedia, Lattice path
Wikipedia, Self-avoiding walk

Cf. A277424, A277756, A284230, A284231, A284461.

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]])(
      `if`(x>y,  b(x-1, y,   0), 0)+
      `if`(y>x,  b(x,   y-1, 0), 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$2, 0)[2]:
seq(a(n), n=0..20);

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]]}][If[x > y,  b[x - 1, y,   0], 0] + If[y > x,  b[x, y - 1, 0], 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, n, 0][[2]];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 19 2017, translated from Maple *)

Recurrence: (768*n^7 - 9760*n^6 + 42960*n^5 - 72624*n^4 + 4272*n^3 + 120634*n^2 - 117042*n + 29523)*a(n) = 4*(1536*n^9 - 17216*n^8 + 56928*n^7 - 19536*n^6 - 199576*n^5 + 257144*n^4 + 67826*n^3 - 200220*n^2 + 46970*n - 201)*a(n-1) - (12288*n^11 - 143872*n^10 + 517376*n^9 - 304896*n^8 - 1803648*n^7 + 3174144*n^6 - 434416*n^5 - 1420224*n^4 - 672608*n^3 + 1216378*n^2 - 69926*n - 51561)*a(n-2) + 8*(n-1)*(3072*n^10 - 40576*n^9 + 179200*n^8 - 212640*n^7 - 583984*n^6 + 1881504*n^5 - 1496616*n^4 - 314158*n^3 + 703776*n^2 - 93829*n - 15912)*a(n-3) - 4*(n-2)*(n-1)*(2*n - 9)*(2*n - 7)*(768*n^7 - 4384*n^6 + 528*n^5 + 22656*n^4 - 24944*n^3 - 2966*n^2 + 8162*n - 1269)*a(n-4). - Vaclav Kotesovec, Apr 25 2017

a(n) ~ c * n^(2*n+4) * 2^(2*n) / exp(2*n), where c = 2.064339567965... - Vaclav Kotesovec, Apr 25 2017

A284461 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) 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, 5, 111, 5127, 400593, 47311677, 7857786015, 1745000283087, 499180661754849, 178734707493557301, 78294815164675006479, 41186656484051421462615, 25619826402721039367943729, 18600984174200732870460447213, 15588291843672510150758754601407

Offset: 0

Alois P. Heinz, Mar 27 2017

Alois P. Heinz, Table of n, a(n) for n = 0..224
Alois P. Heinz, Animation of a(2)=111 walks
Wikipedia, Lattice path
Wikipedia, Self-avoiding walk

Bisection of A284230 (even part).

Cf. A284414, A285673.

b:= proc(n) option remember; `if`(n<2, n+1,
      (n+irem(n, 2))*b(n-1)+(n-1)*b(n-2))
    end:
a:= n-> b(2*n):
seq(a(n), n=0..15);
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 4*n+1,
      ((2*n+1)^2-2)*a(n-1)-(4*n-6)*n*a(n-2))
    end:
seq(a(n), n=0..15);

a[n_] := a[n] = If[n<2, 4n+1, ((2n+1)^2-2) a[n-1] - (4n-6) n a[n-2]];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jun 19 2017, after 2nd Maple program *)

a(n) = A284230(2n).

a(n) = Sum_{k=2n..n*(2n+3)} A284414(2n,k).

A284652 Number T(n,k) of self-avoiding planar walks of length k 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; triangle T(n,k), k>=0, floor((sqrt(1+8*k)-1)/2)<=n<=k, read by columns.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 4, 9, 1, 4, 8, 21, 7, 16, 22, 51, 3, 21, 54, 54, 127, 1, 17, 87, 178, 142, 323, 1, 15, 87, 269, 565, 370, 835, 10, 116, 370, 896, 1766, 983, 2188, 9, 99, 499, 1473, 2776, 5446, 2627, 5798, 4, 91, 536, 2290, 5528, 8657, 16655, 7086, 15511

Offset: 0

Alois P. Heinz, Mar 31 2017

			Triangle T(n,k) begins:
1;
.  1, 1;
.  .  2, 1, 1,  1;
.  .  .  4, 4,  4,  7,   3,   1,   1;
.  .  .  .  9,  8, 16,  21,  17,  15,   10,    9, ... ;
.  .  .  .  .  21, 22,  54,  87,  87,  116,   99, ... ;
.  .  .  .  .   .  51,  54, 178, 269,  370,  499, ... ;
.  .  .  .  .   .   .  127, 142, 565,  896, 1473, ... ;
.  .  .  .  .   .   .    .  323, 370, 1766, 2776, ... ;
.  .  .  .  .   .   .    .    .  835,  983, 5446, ... ;
.  .  .  .  .   .   .    .    .     . 2188, 2627, ... ;

Alois P. Heinz, Columns k = 0..160, flattened
Wikipedia, Lattice path
Wikipedia, Self-avoiding walk

Row sums give A284230.
Column sums give A284415.
Antidiagonal sums give A284428.
T(n,n) gives A001006.
T(n,n+1) gives A284778.
T(n,2n) gives A284416.
T(n,n*(n+1)/2) gives A284418.
Column heights give A122797(k+1).
Cf. A000096, A284231, A284461, A284414 (this triangle read by rows).

Sum_{k=n..n*(n+3)/2} (k+1) * T(n,k) = A284231(n).

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A001900 Successive numerators of Wallis's approximation to Pi/2 (unreduced).

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A277358 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).

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A284461 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) 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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