A100223 G.f. A(x) satisfies: 2^n - 1 = Sum_{k=0..n} [x^k]A(x)^n and also satisfies: (2+z)^n - (1+z)^n + z^n = Sum_{k=0..n} [x^k](A(x)+z*x)^n for all z, where [x^k]A(x)^n denotes the coefficient of x^k in A(x)^n.

1, 0, 1, 1, 0, -2, -3, 1, 11, 15, -13, -77, -86, 144, 595, 495, -1520, -4810, -2485, 15675, 39560, 6290, -159105, -324805, 87075, 1592843, 2616757, -2136539, -15726114, -20247800, 32296693, 152909577, 145139491, -417959049, -1460704685, -885536173, 4997618808, 13658704994, 3223741399

Offset

0,6

Comments

The current sequence and A007440, A214649, A343773 form a cluster. The base of the cluster is A343773. We get the current sequence if we shift all the terms in A343773 to the right by 2 and take a(0) = 1. - Gennady Eremin, May 15 2021

Links

Gennady Eremin, Table of n, a(n) for n = 0..800

Gennady Eremin, Walking in the OEIS: From Motzkin numbers to Fibonacci numbers. The "shadows" of Motzkin numbers, arXiv:2108.10676 [math.CO], 2021.

Formula

G.f.: A(x) = (1+x+sqrt(1-2*x+5*x^2))/2.

G.f.: A(x) = x/(series_reversion[x*(1-x)/(1-x-x^2)]).

D-finite with recurrence a(n) = ((2*n-3)*a(n-1) - 5*(n-3)*a(n-2))/n (for n>2), with a(0)=1, a(1)=0, a(2)=1.

Given g.f. A(x), then B(x) = A(x) - 1 + x series reversion is -B(-x). - Michael Somos, Sep 07 2005

Given g.f. A(x) and C(x) = reversion of x + x^2, then B(x) = A(x) - 1 + x satisfies B(x) = x + C(x * B(x)). - Michael Somos, Sep 07 2005

From Paul Barry, Nov 07 2010: (Start)

a(n+1) has Hankel transform (-1)^n*F(n).

a(n+2) has Hankel transform (-1)^comb(n+1,2).

a(n+3) has Hankel transform (-1)^comb(n+1,2)*F(n+2).

Hankel transform of a(n+4) is sum{k=0..n, (-1)^(n-k+1)*(n-k+1)(F(2k+2)-C(1,k)-C(0,k))}. (End)

G.f.: A(x) = x + G(0) = 1 + x^2/G(0) where G(k) = 1 - x + x^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 25 2011

Lim sup n->infinity |a(n)|^(1/n) = sqrt(5).

From Gennady Eremin, May 15 2021: (Start)

a(n+2) = A343773(n), n >= 0.

G.f.: A(x) = 1 + x^2*B(x), where B(x) is the g.f. of A343773.

Lim_{n->infinity} a(n)/A001006(n) = 0. (End)

Example

G.f. = 1 + x^2 + x^3 - 2*x^5 - 3*x^6 + x^7 + 11*x^8 + 15*x^9 - 13*x^10 - 77*x^11 - 86*x^12 + 144*x^13 + ...
From the table of powers of A(x) (A100224), we see that
2^n-1 = Sum of coefficients [x^0] through [x^n] in A(x)^n:
A^1=[1,0],1,1,0,-2,-3,1,11,...
A^2=[1,0,2],2,1,-2,-5,-2,12,...
A^3=[1,0,3,3],3,0,-5,-6,6,...
A^4=[1,0,4,4,6],4,-2,-8,-3,...
A^5=[1,0,5,5,10,10],5,-5,-10,...
A^6=[1,0,6,6,15,18,17],6,-9,...
A^7=[1,0,7,7,21,28,35,28],7,...
A^8=[1,0,8,8,28,40,60,64,46],...
the main diagonal of which is A001610 = [0,2,3,6,10,17,...],
where Sum_{n>=1} A001610(n-1)/n*x^n = log((1-x)/(1-x-x^2)).

Mathematica

CoefficientList[Series[(1+x+Sqrt[1-2*x+5*x^2])/2, {x, 0, 30}], x] (* Vaclav Kotesovec, Feb 11 2015 *)

Prog

(PARI) {a(n) = if( n<=0, n==0, (2^n - 1 - sum( k=0, n, polcoeff( sum( j=0, min(k, n-1), a(j) * x^j)^n + x * O(x^k), k))) / n)}
(PARI) {a(n) = if( n<=0, n==0, if( n==1, 0, if( n==2, 1, ((2*n-3) * a(n-1) - 5 * (n-3) * a(n-2)) / n)))}
(PARI) {a(n) = if( n<0, 0, polcoeff((1 + x + sqrt(1 - 2*x + 5*x^2 + x * O(x^n))) / 2, n))}
(Python)
A100223 = [1, 0, 1]
for n in range(3, 801):
    A100223.append( ((2*n-3)*A100223[-1]
      - 5*(n-3)*A100223[-2])//n )  # Gennady Eremin, Apr 24 2021

Crossrefs

Cf. A001006, A001610, A007440, A100224, A100225, A214649, A343773.

Sequence in context: A214649 A007440 A343773 * A174017 A178081 A368272

Adjacent sequences: A100220 A100221 A100222 * A100224 A100225 A100226

Keyword

sign

Author

Paul D. Hanna, Nov 28 2004

Extensions

Entry revised by Paul D. Hanna, Mar 19 2013

Status

approved