A277360 -id:A277360 - cp's OEIS frontend

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).

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

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

cp's OEIS Frontend

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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