A047002 -id:A047002 - cp's OEIS frontend

A185286 Triangle T(n,k) is the number of nonnegative walks of n steps with step sizes 1 and 2, starting at 0 and ending at k.

1, 0, 1, 1, 2, 1, 1, 2, 1, 2, 5, 6, 3, 3, 3, 1, 11, 11, 13, 17, 13, 7, 6, 4, 1, 24, 41, 52, 44, 43, 40, 25, 14, 10, 5, 1, 93, 120, 152, 176, 161, 126, 107, 80, 45, 25, 15, 6, 1, 272, 421, 550, 559, 561, 524, 412, 303, 227, 146, 77, 41, 21, 7, 1, 971, 1381, 1813, 2056, 2045, 1835, 1615, 1309, 938, 648, 435, 251, 126, 63, 28, 8, 1

Offset: 0

Franklin T. Adams-Watters, Mar 10 2011

Equivalently, the number of paths from (0,0) to (n,k) using steps of the form (1,2),(1,1),(1,-1) or (1,-2) and staying on or above the x-axis.

It appears that A047002 gives the row sums of this triangle.

			The table starts:
1
0,1,1
2,1,1,2,1
2,5,6,3,3,3,1

Robert Israel, Table of n, a(n) for n = 0..10200

Columns k=0..2 are A187430, A055113, A296619.

Cf. A005408(row lengths), A047002(apparently row sums).

T:= proc(n,k) option remember;
  if k < 0 or k > 2*n then return 0 fi;
  procname(n-1,k-2)+procname(n-1,k-1)+procname(n-1,k+1)+procname(n-1,k+2)
end proc:
T(0,0):= 1:
for nn from 0 to 10 do
  seq(T(nn,k),k=0..2*nn)
od; # Robert Israel, Dec 19 2017

T[n_, k_] := T[n, k] = If[k < 0 || k > 2n, 0, T[n-1, k-2] + T[n-1, k-1] + T[n-1, k+1] + T[n-1, k+2]];
T[0, 0] = 1;
Table[T[n, k], {n, 0, 10}, {k, 0, 2n}] // Flatten (* Jean-François Alcover, Aug 19 2022, after Robert Israel *)

flip(v)=vector(#v,i,v[#v+1-i])
ar(n)={local(p);p=1;
for(k=1,n,p*=1+x+x^3+x^4;p=(p-polcoeff(p,0)-polcoeff(p,1)*x)/x^2);
flip(Vec(p))}

A278416 Number of meanders (walks starting at the origin and ending at any altitude >= 0 that may touch but never go below the x-axis) with n steps from {-4,-3,-2,-1,1,2,3,4}.

Original entry on oeis.org

1, 4, 26, 174, 1231, 8899, 65492, 487646, 3664123, 27723979, 210946444, 1612394958, 12371547879, 95230159650, 735060394986, 5687343753535, 44096482961189, 342530654187820, 2665058975987628, 20765913987073659, 162019898098364055, 1265622208055843635

Offset: 0

Michael Wallner, Nov 21 2016

Andrew Howroyd, Table of n, a(n) for n = 0..200
C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv:1609.06473 [math.CO], 2016.

Cf. A001405, A047002, A278398, A278396, A205337.

seq[n_] := Module[{v = Table[1, n], m = Sum[ x^i, {i, -4, 4}] - 1, p = 1}, For[i = 2, i <= n, i++, p = Expand[p*m]; p = p - Select[p, Exponent[#, x] < 0&]; v[[i]] = (p /. x -> 1)]; v];
seq[25] (* Jean-François Alcover, Jul 11 2018, after Andrew Howroyd *)

seq(n)={my(v=vector(n), m=sum(i=-4, 4, x^i)-1, p=1); v[1]=1; for(i=2, n, p*=m; p-=frac(p); v[i]=subst(p,x,1)); v} \\ Andrew Howroyd, Jun 27 2018

cp's OEIS Frontend

A185286 Triangle T(n,k) is the number of nonnegative walks of n steps with step sizes 1 and 2, starting at 0 and ending at k.

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