A157674 G.f.: A(x) = 1 + x/exp( Sum_{k>=1} (A((-1)^k*x) - 1)^k/k ).

1, 1, 1, -1, -3, 1, 9, 1, -27, -13, 81, 67, -243, -285, 729, 1119, -2187, -4215, 6561, 15505, -19683, -56239, 59049, 202309, -177147, -724499, 531441, 2589521, -1594323, -9254363, 4782969, 33111969, -14348907, -118725597, 43046721

Offset

0,5

Links

Muniru A Asiru, Table of n, a(n) for n = 0..200

Formula

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

a(n) = Sum(k=1..n-1, (k*Sum(j=0..n, j*2^j*(-1)^j*binomial(n,j)* binomial(2*(n-k)-j-1,n-k-1)))/(n*(n-k)))+(-1)^(n-1) n>0, a(0)=1. - Vladimir Kruchinin, Apr 17 2011

a(2*n) = (-3)*a(2*n-2) = (-3)^(n-1), n >= 1; a(2*n+1) = (-3)*a(2*n-1) - 2*(-1)^n*A000108(n-1). - Philippe Deléham, Feb 02 2012

a(2*n+1) = (-1)^n * A137720(n). - Vaclav Kotesovec, Jul 31 2014

Example

G.f.: A(x) = 1 + x + x^2 - x^3 - 3*x^4 + x^5 + 9*x^6 + x^7 - 27*x^8 - ...
ILLUSTRATION OF G.F.:
A(x) = 1 + x/exp((A(-x)-1) + (A(x)-1)^2/2 + (A(-x)-1)^3/3 + (A(x)-1)^4/4 + ...)
RELATED EXPANSION:
Coefficients of 1/A(x) include central binomial coefficients:
[1, -1, 0, 2, 0, -6, 0, 20, 0, -70, 0, 252, 0, -924, ...].
From Philippe Deléham, Feb 02 2012: (Start)
a(2) = 1,
a(4) = (-3)*1 = -3,
a(6) = (-3)*(-3) = 9,
a(8) = (-3)*9 = -27,
a(10) = (-3)*(-27) = 81,
a(12) = (3)*81 = -243, etc.
a(1) = 1,
a(3) = (-3)*1 + 2*1 = -1,
a(5) = (-3)*(-1)- 2*1 = 1,
a(7) = (-3)*1 + 2*2 = 1,
a(9) = (-3)*1 - 2*5 = -13,
a(11) = (-3)*(-13) + 2*14 = 67,
a(13) = (-3)*67 - 2*42 = -285,
a(15) = (-3)*(-285) + 2*132 = 1119, etc. (End)

Maple

1, seq(sum(k*sum(j*2^j*(-1)^j*binomial(n, j)*binomial(2*(n-k)-j-1, n-k-1)/(n*(n-k)), j=0..n), k=1..n-1) +(-1)^(n-1), n=1..200); # Muniru A Asiru, Feb 04 2018

Mathematica

CoefficientList[Series[Sqrt[1+4*x^2]/(Sqrt[1+4*x^2] -x), {x, 0, 40}], x] (* G. C. Greubel, Nov 17 2018 *)

Prog

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+x*exp(-sum(k=1, n, (subst(A, x, (-1)^k*x+x*O(x^n))-1)^k/k))); polcoeff(A, n)}
(Maxima) a(n):=sum((k*sum(j*2^j*(-1)^j*binomial(n, j)*binomial(2*(n-k)-j-1, n-k-1), j, 0, n))/(n*(n-k)), k, 1, n-1)+(-1)^(n-1); /* Vladimir Kruchinin, Apr 17 2011 */
(GAP) Concatenation([1], List([1..200], n->Sum([1..n-1], k->k*Sum([0..n], j->j*2^j*(-1)^j*Binomial(n, j)*Binomial(2*(n-k)-j-1, n-k-1))/(n*(n-k)))+(-1)^(n-1))); # Muniru A Asiru, Feb 04 2018
(Sage) s= (sqrt(1+4*x^2)/(sqrt(1+4*x^2) - x)).series(x, 40); s.coefficients(x, sparse=False) # G. C. Greubel, Nov 17 2018

Crossrefs

Cf. A000108, A000984, A156909, A137720.

Sequence in context: A090261 A303552 A130599 * A360752 A063467 A223140

Adjacent sequences: A157671 A157672 A157673 * A157675 A157676 A157677

Keyword

sign

Author

Paul D. Hanna, Mar 05 2009

Status

approved