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A014307
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Expansion of sqrt( exp(x) / ( 2 - exp(x) )).
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14
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1, 1, 2, 7, 35, 226, 1787, 16717, 180560, 2211181, 30273047, 458186752, 7596317885, 136907048461, 2665084902482, 55726440112987, 1245661569161135, 29642264728189066, 748158516941653967, 19962900431638852297
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Tha Hankel transform of this sequence is A121835. - Philippe DELEHAM, Aug 31 2006
a(n) is the moment of order (n-1) for the discrete measure associated to the weight rho(j+1/2)=2^(j+1/2)/(Pi*binomial(2j+1,j+1/2)), with j integral. So we have a(n)=sum((j+1/2)^(n-1)*rho(j+1/2),j=0..infinity). [From Roland Groux, Jan 05 2009]
Let f(n) = sum(j^n*2^j/binomial(2*j,j),j=1..infinity) = r_n*Pi/2 + s_n; sequence gives r_{n-1}. For example, f(0) through f(5) are [1+1/2*Pi, 3+Pi, 11+7/2*Pi, 55+35/2*Pi, 355+113*Pi, 2807+1787/2*Pi]. For s_n see A180875. - N. J. A. Sloane, following a suggestion from Herb Conn, Feb 08 2011.
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REFERENCES
| M. Klazar, Twelve countings with rooted plane trees, European Journal of Combinatorics 18 (1997), 195-210; Addendum, 18 (1997), 739-740.
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LINKS
| D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions. arXiv:math/0501052v2 [math.CA]
F. J. Dyson, N. E. Frankel and M. L. Glasser, Lehmer's Second Interesting Series, Sep 22, 2010.
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FORMULA
| a(n+1) = 1 + sum(j=1..n, (-1+binomial(n+1,j))*a(n) ) - Jon Perry, Apr 25 2005
The Hankel transform of this sequence is A121835 . - Philippe DELEHAM, Aug 31 2006
E.g.f. A(x) satisfies: A(x) = 1 + integral( A(x)^3 * exp(-x) ). - Paul D. Hanna, Jan 24 2008
a(n) = sum(m=1..n, sum(k=m..n, stirling2(n,k)*k!*binomial(k-1,m-1))/(m)*binomial(2*m-2,m-1)*(-1)^(m-1)/2^(m-1)), e.g.f. B(x) = int {t = 0..x} A(t) satisfies B'(x)=tan(B(x))+sec(B(x)). [From Vladimir Kruchinin, May 10 2011]
From Peter Bala, Aug 25 2011: (Start)
It follows from Kruchinin's formula above that
sum {n = 1..inf} a(n-1)*x^n/n! = series reversion (int {t = 0..x} 1/(sec(t)+tan(t))) = series reversion (int {t = 0..x} (sec(t)-tan(t))) = series reversion (x-x^2/2!+x^3/3!-2*x^4/4!+5*x^5/5!-16*x^6/6!+...) = x+x^2/2!+2*x^3/3!+7*x^4/4!+35*x^5/5!+226*x^6/6!+....
Let f(x) = sec(x)+tan(x). Define the nested derivative D^n[f](x) by means of the recursion D^0[f](x) = 1 and D^(n+1)[f](x) = d/dx(f(x)*D^n[f](x)) for n >= 0 (see A145271 for the coefficients in the expansion of D^n[f](x) in powers of f(x)). Then by [Dominici, Theorem 4.1] we have a(n) = D^n[f](0). Compare with A190392.
(End)
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PROG
| (PARI) {a(n)=n!*polcoeff((exp(x +x*O(x^n))/(2-exp(x +x*O(x^n))))^(1/2), n)} (PARI) /* As solution to integral equation: */ {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+intformal(A^3*exp(-x+x*O(x^n)))); n!*polcoeff(A, n)} - Paul D. Hanna, Jan 24 2008
(Maxima)
a(n):=sum(sum(stirling2(n, k)*k!*binomial(k-1, m-1), k, m, n)/(m)*binomial(2*m-2, m-1)*(-1)^(m-1)/2^(m-1), m, 1, n); [From Vladimir Kruchinin, May 10 2011]
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CROSSREFS
| Cf. A000110, A180875.
Variants: A136727, A136728, A136729. A190392.
Row sums of triangle A156920 (row sums (n) = a(n+1)) [Johannes W. Meijer, Feb 20 2009]
Sequence in context: A201690 A080831 A006947 * A000154 A003713 A058129
Adjacent sequences: A014304 A014305 A014306 * A014308 A014309 A014310
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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