A023426 -id:A023426 - cp's OEIS frontend

A361229 G.f. A(x) satisfies A(x) = 1 + x^4 * (A(x) / (1 - x))^2.

1, 0, 0, 0, 1, 2, 3, 4, 7, 14, 27, 48, 84, 152, 284, 532, 987, 1826, 3401, 6384, 12024, 22656, 42728, 80780, 153151, 290970, 553601, 1054688, 2012373, 3845646, 7359345, 14100692, 27048061, 51941850, 99855389, 192163904, 370159216, 713672568, 1377168108, 2659729380

Offset: 0

Seiichi Manyama, Oct 15 2023

nonn

Partial sums give A023426.

Cf. A002026, A006319, A052702.

A361229 := proc(n)
    add(binomial(n-2*k-1,n-4*k) * binomial(2*k,k) / (k+1),k=0..floor(n/4)) ;
end proc:
seq(A361229(n),n=0..70) ; # R. J. Mathar, Dec 04 2023

a(n) = sum(k=0, n\4, binomial(n-2*k-1, n-4*k)*binomial(2*k, k)/(k+1));

G.f.: A(x) = 2*(1-x) / (1-x+sqrt((1-x)^2-4*x^4)).
a(n) = Sum_{k=0..floor(n/4)} binomial(n-2*k-1,n-4*k) * binomial(2*k,k) / (k+1).
D-finite with recurrence (n+4)*a(n) +(-3*n-7)*a(n-1) +(3*n+2)*a(n-2) +(-n+1)*a(n-3) +4*(-n+2)*a(n-4) +4*(n-4)*a(n-5)=0. - R. J. Mathar, Dec 04 2023

A374585 A family of marked Motzkin-like paths.

Original entry on oeis.org

1, 1, 1, 1, 3, 9, 21, 41, 79, 169, 393, 913, 2051, 4537, 10165, 23257, 53759, 124153, 286009, 660161, 1531875, 3571753, 8348981, 19539209, 45792719, 107546633, 253153609, 597034609, 1410131683, 3334984025, 7897992213, 18730123449, 44476842431, 105740699609, 251664629689

Offset: 0

Emanuele Munarini, Jul 12 2024

Number of lattice paths from (0,0) to (n,0) that stay weakly in the first quadrant and such that each step is either H=(1,0), U=(2,1) or D=(2,-1), where the steps H and D can be marked H*, D*, so that (in the canonical decomposition) a marked step H* cannot be followed by the empty path or by H. For instance, a(5)=9 because we have HHHHH, HUD, HUD*, H*UD, H*UD*, UDH, UD*H, UHD and UHD*.

Cf. A023426.

CoefficientList[Series[((1-t)^2-Sqrt[1-4t+6t^2-4t^3-7t^4+8t^5])/(4t^4),{t,0,100}],t]
Table[Sum[Binomial[n-k,3k] 2^k CatalanNumber[k], {k,0,n/4}], {n,0,100}]

G.f: A(t) = ((1-t)^2-sqrt(1-4*t+6*t^2-4*t^3-7*t^4+8*t^5))/(4*t^4).
G.f. A(x) satisfies: A(t) = 1+t*A(t)+t*(A(t)-1-t*A(t))+2*t^4*A(t)^2, or 2*t^4*A(t)^2-(1-t)^2*A(t)+1-t = 0.
a(n) = Sum_{k=0..floor(n/4)} binomial(n-k,3*k)*2^k*Catalan(k).
Recurrence: a(n+4) = 2*a(n+3) - a(n+2) + 2*Sum_{k=0..n} a(k)*a(n-k).
D-finite with recurrence: (n+9)*a(n+5) -2*(2*n+15)*a(n+4) +6*(n+6)*a(n+3) -2*(2*n+9)*a(n+2) -7*(n+3)*a(n+1) +4*(2*n+3)*a(n)=0.
a(n) ~ (1/2)*sqrt((4-X)/(2*Pi))*X^(-n-2)/n^(3/2),
where X = 0.40355658567370456... is the unique positive real root of 8*x^3-7*x^2+4*x-1.

cp's OEIS Frontend

A361229 G.f. A(x) satisfies A(x) = 1 + x^4 * (A(x) / (1 - x))^2.

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