A201641 -id:A201641 - cp's OEIS frontend

A330799 Evaluation of the Motzkin polynomials at 1/2 and normalized with 2^n.

1, 3, 13, 59, 285, 1419, 7245, 37659, 198589, 1059371, 5705517, 30976571, 169338781, 931239243, 5147825421, 28587660123, 159406327677, 892113040491, 5009160335085, 28210229053563, 159304938535773, 901845743050635, 5117144607546573, 29096321095698843, 165765778648482621

Offset: 0

Peter Luschny, Jan 01 2020

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A064189, A129400, A201641, A330800.

m:=30;
R:=PowerSeriesRing(Rationals(), m+2);
A330799:= func< n | Coefficient(R!( 2/(1-4*x+Sqrt((1-6*x)*(1+2*x))) ), n) >;
[A330799(n): n in [0..m]]; // G. C. Greubel, Sep 14 2023

a := proc(n) option remember; if n < 3 then return [1, 3, 13][n+1] fi;
(-84*(n - 2)*a(n-3) - 2*(8*n + 5)*a(n-2) + (11*n + 5)*a(n-1))/(n + 1) end:
seq(a(n), n=0..24);
# Alternative:
gf := 2/(1 - 4*x + sqrt((1 - 6*x)*(2*x + 1))):
ser := series(gf, x, 30): seq(coeff(ser,x,n), n=0..24);
# Or:
series((x^2+x)/(7*x^2+4*x+1), x, 30): gfun:-seriestoseries(%, 'revogf'):
convert(%, polynom) / x: seq(coeff(%, x, n), n=0..24);

With[{C = Binomial}, A064189[n_, k_] := Sum[C[n, j]* (C[n-j, j+k] - C[n-j, j+k+2]), {j, 0, n}]];
a[n_] := 2^n*Sum[A064189[n, k]/2^k, {k, 0, n}];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Sep 25 2022 *)
(* Second program *)
A330799[n_]:= Coefficient[Series[2/(1-4*x+Sqrt[(1-6*x)*(1+2*x)]), {x, 0,50}], x, n]; Table[A330799[n], {n,0,30}] (* G. C. Greubel, Sep 14 2023 *)

R. = PowerSeriesRing(QQ)
f = (x^2 + x)/(7*x^2 + 4*x+1)
f.reverse().shift(-1).list()

a(n) = Sum_{k=0..n} A201641(n,k).
a(n) = 2^n*Sum_{k=0..n} A064189(n,k)/2^k.
a(n) = (-84*(n - 2)*a(n-3) - 2*(8*n + 5)*a(n-2) + (11*n + 5)*a(n-1))/(n + 1).
a(n) = [x^n] 2/(1 - 4*x + sqrt((1 - 6*x)*(2*x + 1))).
a(n) = [x^n] reverse((x^2 + x)/(7*x^2 + 4*x+1))/x.

A330800 Evaluation of the Motzkin polynomials at -1/2 and normalized with (-2)^n.

Original entry on oeis.org

1, -1, 5, -17, 77, -345, 1653, -8097, 40733, -208553, 1084421, -5708785, 30370861, -163019641, 881790357, -4801746753, 26302052925, -144825094473, 801155664933, -4450426297233, 24815385947469, -138842668857369, 779247587235765, -4385948395419873, 24750623835149661

Offset: 0

Peter Luschny, Jan 01 2020

sign

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A064189, A129400, A201641, A330799.

I:=[1,-1,5]; [n le 3 select I[n] else ((6-n)*Self(n-1) + 6*(4*n-9)*Self(n-2) -36*(n-3)*Self(n-3))/n: n in [1..30]]; // G. C. Greubel, Sep 13 2023

a := proc(n) option remember; if n < 3 then return [1, -1, 5][n+1] fi;
(-36*(n - 2)*a(n-3) + 6*(4*n - 5)*a(n-2) - (n - 5)*a(n-1))/(n + 1) end:
seq(a(n), n=0..24);
# Alternative:
gf := 2/(sqrt(4*x - 12*x^2 + 1) + 1):
ser := series(gf, x, 30): seq(coeff(ser,x,n), n=0..24);
# Or:
series((x^2+x)/(3*x^2+1), x, 30): gfun:-seriestoseries(%, 'revogf'):
convert(%, polynom) / x: seq(coeff(%, x, n), n=0..24);

A330800[n_]:= Coefficient[Series[2/(Sqrt[4*x-12*x^2+1] +1), {x,0,50}], x, n]; Table[A330800[n], {n, 0, 30}] (* G. C. Greubel, Sep 13 2023 *)

R. = PowerSeriesRing(QQ)
f = (x^2 + x)/(3*x^2 + 1)
f.reverse().shift(-1).list()

a(n) = Sum_{k=0..n} (-1)^(n-k)*A201641(n,k).
a(n) = (-2)^n*Sum_{k=0..n} A064189(n,k)/(-2)^k.
a(n) = (-36*(n-2)*a(n-3) + 6*(4*n-5)*a(n-2) - (n-5)*a(n-1))/(n+1).
a(n) = [x^n] 2/(sqrt(4*x - 12*x^2 + 1) + 1).
a(n) = [x^n] reverse((x^2 + x)/(3*x^2 + 1))/x.

cp's OEIS Frontend

A330799 Evaluation of the Motzkin polynomials at 1/2 and normalized with 2^n.

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