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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 - acosh(sqrt(3/2))).

E.g.f. for the same sequence, but with inverted signs of even terms: -sqrt(3)*tanh(z*sqrt(3)/2 + acosh(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 * A073910 A115251 A057241

Adjacent sequences:  A289061 A289062 A289063 * A289065 A289066 A289067

KEYWORD

sign

AUTHOR

Stanislav Sykora, Jun 23 2017

STATUS

approved

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Last modified September 26 10:52 EDT 2017. Contains 292518 sequences.