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
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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 *)
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
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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 *)
A277176
Exponential convolution of Catalan numbers and factorial numbers.
Original entry on oeis.org
1, 2, 6, 23, 106, 572, 3564, 25377, 204446, 1844876, 18465556, 203179902, 2438366836, 31699511768, 443795839192, 6656947282725, 106511191881270, 1810690391626380, 32592427526913540, 619256124778620450, 12385122502136529420, 260087572569333384840
Offset: 0
-
a:= proc(n) option remember; `if`(n<2, n+1,
((n^2+5*n-2)*a(n-1)-(4*n-2)*(n-1)*a(n-2))/(n+1))
end:
seq(a(n), n=0..30);
-
a[n_] := Sum[Binomial[n, i] CatalanNumber[i] (n-i)!, {i, 0, n}];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 21 2020 *)
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
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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 *)
A277756
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) 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, 5, 32, 224, 1723, 14569, 135286, 1375882, 15263414, 183817326, 2391291386, 33443618930, 500611975023, 7988044467121, 135376576319870, 2428721569276698, 45988428905194350, 916607431346170686, 19182997480530342168, 420606731490047403144
Offset: 0
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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`(y=x, b(x-1, y-1, 0), 0)+
`if`(y>x+1 and t<>2, b(x+1, y-1, 1), 0)+
`if`(y>=x and t<>1, b(x-1, y+1, 2), 0))))
end:
a:= n-> b(n$2, 0)[2]:
seq(a(n), n=0..25);
-
b[x_, y_, t_] := b[x, y, t] = If[x < 0 || y < 0, {0, 0}, If[x == 0 && y == 0, {1, 1}, # + {0, #[[1]]}&[If[y < x, b[x-1, y, 0], 0] + If[y <= x, b[x, y-1, 0], 0] + If[y >= x, b[x-1, y-1, 0], 0] + If[y > x+1 && t != 2, b[x+1, y-1, 1], 0] + If[y >= x && t != 1, b[x-1, y+1, 2], 0]]]];
a[n_] := b[n, n, 0][[2]];
a /@ Range[0, 25] (* Jean-François Alcover, Oct 19 2019, after Alois P. Heinz *)
Showing 1-5 of 5 results.
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