A176675 Sequence defined by the recurrence formula a(n+1)=sum(a(p)*a(n-p)+k,p=0..n)+l for n>=1, with here a(0)=1, a(1)=0, k=0 and l=-1.

1, 0, -1, -3, -7, -14, -23, -24, 21, 220, 821, 2261, 4935, 7814, 3615, -34251, -179511, -593420, -1521779, -3001089, -3500101, 4410846, 44902907, 179440150, 526896835, 1212055740, 1976482795, 697522595, -10858119895, -56563343774

Offset

0,4

Links

Table of n, a(n) for n=0..29.

Formula

G.f.: f(z) = (1-sqrt(1-4*z*(a(0)-z*a(0)^2+z*a(1)+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z) (k=0, l=-1).

a(n) = sum(k=0..n, (A000108(k) * sum(i=0..n-k, binomial(k+1,n-k-i)*binomial(k+i,k)*(-2)^(n-k-i)))), where A000108 is the Catalan numbers. [Vladimir Kruchinin, Nov 11 2012]

Conjecture: +(n+1)*a(n) +2*(-3*n+1)*a(n-1) +(13*n-21)*a(n-2) +4*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Feb 29 2016

Example

a(2)=2*0*1-1=-1. a(2)=2*1*(-1)+0^2-1=-3. a(4)=2*1*(-3)+2*0*(-1)-1=-7.

Maple

l:=-1: : k := 0 : m:=0: d(0):=1:d(1):=m: for n from 1 to 30 do d(n+1):=sum(d(p)*d(n-p)+k, p=0..n)+l:od :
taylor((1-sqrt(1-4*z*(d(0)-z*d(0)^2+z*m+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z), z=0, 30); seq(d(n), n=0..30);

Prog

(Maxima) a(n):=sum((binomial(2*k, k)*sum(binomial(k+1, n-k-i)*binomial(k+i, k)*(-2)^(n-k-i), i, 0, n-k))/(k+1), k, 0, n); /* Vladimir Kruchinin, Nov 16 2012 */
(PARI) /* Using Vladimir Kruchinin's formula: */
{A000108(k)=binomial(2*k, k)/(k+1)}
{a(n)=sum(k=0, n, (A000108(k)*sum(i=0, n-k, binomial(k+1, n-k-i)*binomial(k+i, k)*(-2)^(n-k-i))))} \\ Paul D. Hanna, Nov 15 2012

Crossrefs

Cf. A176648.

Sequence in context: A173209 A331240 A146931 * A377122 A213077 A294400

Adjacent sequences: A176672 A176673 A176674 * A176676 A176677 A176678

Keyword

easy,sign

Author

Richard Choulet, Apr 23 2010

Status

approved