A289068 Recurrence a(n+2) = Sum_{k=0..n} binomial(n,k)*a(k)*a(n+1-k) with a(0)=1, a(1)=-2.

1, -2, -2, 2, 14, 10, -170, -670, 2270, 30490, 26950, -1435150, -8513650, 59564650, 1050090550, 486517250, -113618013250, -831340535750, 10136160835750, 208459859695250, -121723298991250, -41568491959973750, -338549875950886250, 6637158567781561250

Offset

0,2

Comments

One of a family of integer sequences whose e.g.f.s satisfy the differential equation f''(z) = f'(z)f(z). For more details, see A289064.

Links

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

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

Formula

E.g.f.: -sqrt(5)*tanh(z*sqrt(5)/2 - arccosh(sqrt(5)/2)).

E.g.f. for the sequence (-1)^(n+1)*a(n): -sqrt(5)*tanh(z*sqrt(5)/2 + arccosh(sqrt(5)/2)).

Prog

(PARI) c0=1; c1=-2; nmax = 200;
  a=vector(nmax+1); a[1]=c0; a[2]=c1;
  for(m=0, #a-3, a[m+3]=sum(k=0, m, binomial(m, k)*a[k+1]*a[m+2-k]));
  a
(Python)
from sympy import binomial
l=[1, -2]
for n in range(2, 51): l+=[sum([binomial(n - 2, k)*l[k]*l[n - 1 - k] for k in range(n - 1)]), ]
print(l) # Indranil Ghosh, Jun 30 2017

Crossrefs

Sequences for other starting pairs: A000111 (1,1), A289064 (1,-1), A289065 (2,-1), A289066 (3,1), A289067 (3,-1), A289069 (3,-2), A289070 (0,3).

Sequence in context: A049148 A369077 A183584 * A063898 A074052 A350599

Adjacent sequences: A289065 A289066 A289067 * A289069 A289070 A289071

Keyword

sign

Author

Stanislav Sykora, Jun 23 2017

Status

approved