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Search: id:A155585
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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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8
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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; listen)
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OFFSET
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0,4
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COMMENT
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Factorials have a similar recurrence: f(n) = Sum_{k=0..n-1} C(n-1,k)*f(n-1-k)*f(k), n>0.
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FORMULA
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E.g.f.: A(x) = exp(x)sech(x) [from Paul Barry - see A009006].
Unsigned, equals A009006 (expansion of 1+tan(x)).
a(n) = 2^n E_{n}(1) where E_{n}(x) are the Euler polynomials. [From Peter Luschny (peter(AT)luschny.de), Jan 26 2009]
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 (peter(AT)luschny.de), Apr 22 2009]
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EXAMPLE
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E.g.f.: A(x) = 1 + x - 2*x^3/3! + 16*x^5/5! - 272*x^7/7! + 7936*x^9/9! -+...
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MAPLE
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A155585 := n -> 2^n*euler(n, 1): [From Peter Luschny (peter(AT)luschny.de), Jan 26 2009]
restart: G(x):=exp(x)* sech(x): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=0..26 ); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 05 2009]
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MATHEMATICA
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a[m_] := Sum[(-2)^(m - k) k! StirlingS2[m, k], {k, 0, m}] [From Peter Luschny (peter(AT)luschny.de), Apr 29 2009]
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PROGRAM
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(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)}
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CROSSREFS
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Cf. A009006.
Sequence in context: A146558 A025600 A009006 this_sequence A057375 A009045 A060313
Adjacent sequences: A155582 A155583 A155584 this_sequence A155586 A155587 A155588
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KEYWORD
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sign
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Jan 24 2009
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