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A155585
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a(n) = Sum_{k=0..n-1} (-1)^(k)*C(n-1,k)*a(n-1-k)*a(k) for n>0 with a(0)=1.
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12
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1, 1, 0, -2, 0, 16, 0, -272, 0, 7936, 0, -353792, 0, 22368256, 0, -1903757312, 0, 209865342976, 0, -29088885112832, 0, 4951498053124096, 0, -1015423886506852352, 0, 246921480190207983616, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Factorials have a similar recurrence: f(n) = Sum_{k=0..n-1} C(n-1,k)*f(n-1-k)*f(k), n>0.
Related to A102573: letting T(q,r) be the coefficient of n^(r+1) in the polynomial 2^(q-n)/n times sum(k=0..n binomial(n,k)*k^q ), then A155585(x)= sum(k=0..x-1 T(x,k)*(-1)^k). See Mathematica code below. [From John M. Campbell, Nov 16 2011]
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REFERENCES
| H. M. Edwards, A normal form for elliptic curves, Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 3, 393-422. see section 12, pp. 410-411.
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FORMULA
| E.g.f.: exp(x)*sech(x) = exp(x)/cosh(x) [from Paul Barry - see A009006].
Sequence of absolute values is A009006 (e.g.f. 1+tan(x)).
O.g.f.: Sum_{n>=0} n! * x^n / Product_{k=1..n} (1 + 2*k*x). [From Paul D. Hanna, Jul 20 2011]
a(n) = 2^n*E_{n}(1) where E_{n}(x) are the Euler polynomials. [From Peter Luschny, Jan 26 2009]
a(n) = EL_{n}(-1) where EL_{n}(x) are the Eulerian polynomials. [From Peter Luschny, Aug 03 2010]
a(n) = (4^n-2^n)*B_n(1)/n, where B_{n}(x) are the Bernoulli polynomials (B_n(1) = B_n for n <> 1). [From Peter Luschny, Apr 22 2009]
G.f.: 1/(1-x+x^2/(1-x+4*x^2/(1-x+9*x^2/(1-x+16*x^2/(1-...))))) (continued fraction). [From Paul Barry, Mar 30 2010]
G.f.: -log(x/(exp(x)-1))/x = sum(n>=0, a(n)*x^n/(2^(n+1)*(2^(n+1)-1)*n!) [From Vladimir Kruchinin, Nov 05 2011]
E.g.f.: exp(x)/cosh(x) = 2/(1+exp(-2*x)) = 2/(G(0) + 1) ; G(k) = 1 - 2*x/(2*k + 1 - x*(2*k+1)/(x - (k+1)/G(k+1))) ; (continued fraction). - Sergei N. Gladkovskii, Dec 10 2011
E.g.f. is x(t,1) + y(t,1) where x(t,a) and y(t,a) satisfy y(t,a)^2 = (a^2 - x(t,a)^2) / (1 - a^2 * x(t,a)^2) and dx(t,a) / dt = y(t,a) * (1 - a * x(t,a)^2) and are the elliptic functions of Edwards. - Michael Somos, Jan 16 2012
E.g.f.: 1/(1 - x/(1+x/(1 - x/(3+x/(1 - x/(5+x/(1 - x/(7+x/(1 - x/(9+x/(1 +...))))))))))), a continued fraction. [From Paul D. Hanna, Feb 11 2012]
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EXAMPLE
| E.g.f.: 1 + x - 2*x^3/3! + 16*x^5/5! - 272*x^7/7! + 7936*x^9/9! -+... = exp(x)/cosh(x).
O.g.f.: 1 + x - 2*x^3 + 16*x^5 - 272*x^7 + 7936*x^9 - 353792*x^11 +-...
O.g.f.: 1 + x/(1+2*x) + 2!*x^2/((1+2*x)*(1+4*x)) + 3!*x^3/((1+2*x)*(1+4*x)*(1+6*x)) +...
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MAPLE
| A155585 := n -> 2^n*euler(n, 1): [From Peter Luschny, Jan 26 2009]
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MATHEMATICA
| a[m_] := Sum[(-2)^(m - k) k! StirlingS2[m, k], {k, 0, m}] [From Peter Luschny, Apr 29 2009]
poly[q_] := 2^(q-n)/n*FunctionExpand[Sum[Binomial[n, k]*k^q, {k, 0, n}]]; T[q_, r_] := First[Take[CoefficientList[poly[q], n], {r+1, r+1}]]; Table[Sum[T[x, k]*(-1)^k, {k, 0, x-1}], {x, 1, 16}] (* From John M. Campbell, Nov 16 2011 *)
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PROG
| (PARI) {a(n)=if(n==0, 1, sum(k=0, n-1, (-1)^(k)*binomial(n-1, k)*a(n-1-k)*a(k)))}
(PARI) {a(n)=local(X=x+x*O(x^n)); n!*polcoeff(exp(X)/cosh(X), n)}
(PARI) {a(n)=polcoeff(sum(m=0, n, m!*x^m/prod(k=1, m, 1+2*k*x+x*O(x^n))), n)} /* Paul D. Hanna, Jul 20 2011 */
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); n! * polcoeff( 1 + sinh(x + A) / cosh(x + A), n))} /* Michael Somos, Jan 16 2012 */
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CROSSREFS
| Cf. A009006, A102573.
Equals row sums of A119879. [From Johannes W. Meijer, Apr 20 2011]
Sequence in context: A146558 A025600 A009006 * A057375 A009045 A060313
Adjacent sequences: A155582 A155583 A155584 * A155586 A155587 A155588
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KEYWORD
| sign,changed
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Jan 24 2009
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