login
This site is supported by donations to The OEIS Foundation.

 

Logo

The October issue of the Notices of the Amer. Math. Soc. has an article about the OEIS.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A289064 Recurrence a(n+2) = Sum_{k=0..n} binomial(n,k)*a(k)*a(n+1-k) with a(0)=1, a(1)=-1. 15
1, -1, -1, 0, 3, 6, -9, -90, -153, 1134, 8019, 2430, -262197, -1438074, 4421871, 104152230, 380788047, -4779057186, -63944168661, -55095931890, 5848795071603, 54270718742646, -229189662998649, -9171963685125450, -53834845287495753, 893621501807183694 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

One of a family of integer sequences whose e.g.f.s satisfy the differential equation f''(z) = f'(z)f(z). Each such sequence is uniquely characterized by its two starting terms. When the first term changes sign, the effect is the inversion of the signs of all even terms, leaving all absolute values intact. There are many related sequences in the OEIS (see the link). Starting with a(0) = a(1) = 1, for example, one obtains A000111. All such sequences have a well defined, explicit e.g.f. (see the link).

LINKS

Stanislav Sykora, Table of n, a(n) for n = 0..200

S. Sykora, Sequences related to the differential equation f'' = af'f, Stan's Library, Vol. VI, Jun 2017.

FORMULA

E.g.f.: -sqrt(3)*tanh(z*sqrt(3)/2 - arccosh(sqrt(3/2))).

E.g.f. for the same sequence, but with inverted signs of even terms: -sqrt(3)*tanh(z*sqrt(3)/2 + arccosh(sqrt(3/2))).

MATHEMATICA

a[n_] := a[n] = Sum[Binomial[n-2, k]*a[k]*a[n-k-1], {k, 0, n-2}]; a[0] = 1; a[1] = -1; Array[a, 26, 0] (* Jean-Fran├žois Alcover, Jul 20 2017 *)

PROG

(PARI) c0=1; c1=-1; nmax = 200; \\ Initialize

  a=vector(nmax+1)); a[1]=c0; a[2]=c1; \\ Compute

  for(m=0, #a-3, a[m+3]=sum(k=0, m, binomial(m, k)*a[k+1]*a[m+2-k]));

  a \\ Display

CROSSREFS

Sequences for other starting pairs: A000111 (1,1), A289065 (2,-1), A289066 (3,1), A289067 (3,-1), A289068 (1,-2), A289069 (3,-2), A289070 (0,3).

Sequence in context: A133195 A196156 A103978 * A293537 A073910 A115251

Adjacent sequences:  A289061 A289062 A289063 * A289065 A289066 A289067

KEYWORD

sign

AUTHOR

Stanislav Sykora, Jun 23 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 24 16:53 EDT 2018. Contains 315347 sequences. (Running on oeis4.)