A176606 -id:A176606

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A176607

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)=4, k=0 and l=1.

+10
1

1, 4, 9, 35, 143, 648, 3071, 15126, 76495, 395086, 2074699, 11044027, 59457897, 323180520, 1771081641, 9774955015, 54286011887, 303138215322, 1701016909235, 9586701364893, 54241695455421, 307991483403216, 1754468491846461

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..22.

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).

Conjecture: (n+1)*a(n) +2*(-3*n+1)*a(n-1) +(-3*n+11)*a(n-2) +16*(n-3)*a(n-3) +8*(-n+4)*a(n-4)=0. - R. J. Mathar, Feb 29 2016

a(n) = Sum_{k=0..n}((C(k)*Sum_{l=0..n-k}(binomial(k+1,l)*2^l*binomial(n-2*l,n-l-k)))), where C(k) is a Catalan number (A000108). - Vladimir Kruchinin, Mar 15 2016

EXAMPLE

a(2)=2*1*4+1+9. a(3)=2*1*9+4^2+1=35.

MAPLE

l:=1: : k := 0 : m:=4: d(0):=1:d(1):=m: for n from 1 to 28 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, 31); seq(d(n), n=0..29) ;

PROG

(Maxima)

a(n):=sum((binomial(2*k, k)/(k+1)*sum(binomial(k+1, l)*2^l*binomial(n-2*l, n-l-k), l, 0, n-k)), k, 0, n); /* Vladimir Kruchinin, Mar 15 2016 */

CROSSREFS

Cf. A000108, A176604, A176605, A176606.

KEYWORD

easy,nonn

AUTHOR

Richard Choulet, Apr 21 2010

STATUS

approved

A176609

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)=5, k=0 and l=1.

+10
1

1, 5, 11, 48, 207, 1016, 5159, 27337, 148489, 824232, 4650657, 26602827, 153900879, 898909266, 5293577451, 31395570786, 187364023083, 1124308178270, 6779554362911, 41059231942321, 249646266800185, 1523286825246798

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..21.

FORMULA

G.f 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).

Conjecture: (n+1)*a(n) +2*(-3*n+1)*a(n-1) +(-7*n+19)*a(n-2) +24*(n-3)*a(n-3) +12*(-n+4)*a(n-4)=0. - R. J. Mathar, Mar 01 2016

EXAMPLE

a(2)=2*1*5+1=11. a(3)=2*1*11+5^2+1=48.

MAPLE

l:=1: : k := 0 : m :=5: d(0):=1:d(1):=m: for n from 1 to 28 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, 31); seq(d(n), n=0..29);