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A157674
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G.f.: A(x) = 1 + x/exp( Sum_{k>=1} [A((-1)^k*x) - 1]^k/k ).
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1
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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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. [From Vladimir Kruchinin (kru(AT)tusur.ru), 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). - DELEHAM Philippe, Feb 02 2012
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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 DELEHAM Philippe, 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)
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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); [From Vladimir Kruchinin (kru(AT)tusur.ru), Apr 17 2011]
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CROSSREFS
| Cf. A000108, A000984, A156909.
Sequence in context: A070894 A090261 A130599 * A063467 A021762 A019736
Adjacent sequences: A157671 A157672 A157673 * A157675 A157676 A157677
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KEYWORD
| sign,changed
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Mar 05 2009
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