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
Keywords
Examples
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)
Links
- Muniru A Asiru, Table of n, a(n) for n = 0..200
Programs
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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
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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
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Mathematica
CoefficientList[Series[Sqrt[1+4*x^2]/(Sqrt[1+4*x^2] -x), {x, 0, 40}], x] (* G. C. Greubel, Nov 17 2018 *) -
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 */
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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)} -
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
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