A177111 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)=1, k=-2 and l=0.

1, 1, -2, -9, -30, -84, -204, -389, -326, 1780, 13156, 57452, 197552, 551846, 1138832, 752911, -8109806, -57353648, -255573404, -898715548, -2539157248, -5106161134, -1629827488, 50275158584, 330772150256, 1453122571658

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..1767

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=-2, l=0).

Conjecture: +(n+1)*a(n) +(-7*n+2)*a(n-1) +3*(5*n-7)*a(n-2) +3*(n-8)*a(n-3) +2*(-10*n+41)*a(n-4) +8*(n-5)*a(n-5)=0. - R. J. Mathar, Mar 02 2016

Conjectured recurrence follows from the differential equation (8*z^6 - 20*z^5 + 3*z^4 + 15*z^3 - 7*z^2 + z) * f'(z) + (2*z^4 - 15*z^3 + 9*z^2 - 5*z + 1) * f(z) + 4*z^4 - 11*z^3 + 9*z^2 + 3*z - 1 = 0 satisfied by the g.f. - Robert Israel, Jan 03 2024

Example

a(2)=2*1*1-4=-2. a(3)=2*1*(-2)-4+1^2-2=-9. a(4)=2*1*(-9)-4+2*1*(-2)-4=-30.

Maple

l:=0: : k := -2 : m:=1: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);

Crossrefs

Cf. A177110.

Sequence in context: A352405 A372152 A056778 * A290746 A268586 A056288

Adjacent sequences: A177108 A177109 A177110 * A177112 A177113 A177114

Keyword

easy,sign

Author

Richard Choulet, May 03 2010

Status

approved