A337499 a(n) is the number of ballot sequences of length n tied or won by at most 2 votes.

1, 2, 4, 6, 14, 20, 50, 70, 182, 252, 672, 924, 2508, 3432, 9438, 12870, 35750, 48620, 136136, 184756, 520676, 705432, 1998724, 2704156, 7696444, 10400600, 29716000, 40116600, 115000920, 155117520, 445962870

Offset: 0

Author

Nachum Dershowitz, Aug 29 2020

Keywords

• nonn
• walk

Comments

Also the number of n-step walks on a path graph ending within 2 steps of the origin. Also the number of monotonic paths of length n ending within 2 steps of the diagonal.

Links

• Robert Israel, Table of n, a(n) for n = 0..3323

Crossrefs

Cf. A001791, A128014.

Bisections give A000984 (odd part, starting from second element), A051924 (even part).

Programs

Maple

f:= gfun:-rectoproc({(4 + 4*n)*a(n) + (-12 - 4*n)*a(1 + n) + (-22 - 5*n)*a(2 + n) + (n + 4)*a(n + 3) + (6 + n)*a(n + 4), a(0) = 1, a(1) = 2, a(2) = 4, a(3) = 6},a(n),remember):
map(f, [$0..100]); # Robert Israel, Oct 08 2020

Formula

a(n) = A128014(n+1) + ((n+1) mod 2)*2*A001791(ceiling(n/2)).

D-finite with recurrence +(n+2)*a(n) +n*a(n-1) +(-5*n-2)*a(n-2) +4*(-n+1)*a(n-3) +4*(n-3)*a(n-4)=0. - Conjectured by R. J. Mathar, Sep 27 2020, verified by Robert Israel, Oct 08 2020

G.f.: ((4*x + 2)*sqrt(-4*x^2 + 1) + 4*x^2 + 4*x + 2)/(sqrt(-4*x^2 + 1)*(1 + sqrt(-4*x^2 + 1))^2). - Robert Israel, Oct 08 2020

a(n) ~ 2^(n - 1/2) * (5 + (-1)^n) / sqrt(Pi*n). - Vaclav Kotesovec, Mar 08 2023