A277176 -id:A277176 - cp's OEIS frontend

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.

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

A277175 Convolution of Catalan numbers and factorial numbers.

Original entry on oeis.org

1, 2, 5, 15, 53, 222, 1120, 6849, 50111, 427510, 4142900, 44693782, 529276962, 6813205468, 94642629984, 1410507388421, 22445134308123, 379776665469030, 6808016435182620, 128886547350655050, 2569493300908367550, 53805226930896987540, 1180673761078007109840

Offset: 0

Alois P. Heinz, Oct 02 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..449

Cf. A000108, A000142, A277176, A277359, A277396.

a:= proc(n) option remember; `if`(n<4, [1, 2, 5, 15][n+1],
      ((2*(n^4-n^3-19*n^2+48*n-5))*a(n-1)
       -(n+1)*(n^4+9*n^3-90*n^2+226*n-160)*a(n-2)
       +(2*(4*n^5-18*n^4-23*n^3+266*n^2-523*n+330))*a(n-3)
       -(4*(n-2))*(n^2-4*n+5)*(2*n-5)^2*a(n-4))/
       ((n+1)*(n^2-6*n+10)))
    end:
seq(a(n), n=0..30);

Table[Sum[CatalanNumber[k]*(n - k)!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 13 2016 *)

a(n) = Sum_{i=0..n} C(i) * (n-i)!.

a(n) ~ n! * (1 + 1/n + 2/n^2 + 7/n^3 + 31/n^4 + 163/n^5 + 979/n^6 + 6556/n^7 + 48150/n^8 + 383219/n^9 + 3275121/n^10 + ...), for coefficients see A277396. - Vaclav Kotesovec, Oct 13 2016

cp's OEIS Frontend

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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