The On-Line Encyclopedia of Integer Sequences (OEIS)

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A176606

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)=3, k=0 and l=1.
(history; published version)

#8 by N. J. A. Sloane at Fri Apr 15 13:47:03 EDT 2016

STATUS

proposed

approved

#7 by Vladimir Kruchinin at Fri Apr 15 11:05:39 EDT 2016

STATUS

editing

proposed

#6 by Vladimir Kruchinin at Fri Apr 15 11:05:34 EDT 2016

FORMULA

a(n) = Sum_{k=0..n} (C(k)*Sum_{j=0..n-k} (binomial(k+1,n-k-j)*binomial(-n+ 2*k+2*j,j))). - Vladimir Kruchinin, Apr 15 2016

PROG

(Maxima)

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

STATUS

approved

editing

#5 by R. J. Mathar at Mon Feb 29 14:19:59 EST 2016

STATUS

proposed

approved

#4 by R. J. Mathar at Mon Feb 29 13:33:57 EST 2016

STATUS

editing

proposed

#3 by R. J. Mathar at Mon Feb 29 13:33:53 EST 2016

FORMULA

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

STATUS

approved

editing

#2 by Charles R Greathouse IV at Wed Dec 19 15:50:23 EST 2012

AUTHOR

_Richard Choulet (richardchoulet(AT)yahoo.fr), _, Apr 21 2010

Discussion

Wed Dec 19

15:50

OEIS Server: https://oeis.org/edit/global/1844

#1 by N. J. A. Sloane at Tue Jun 01 03:00:00 EDT 2010

NAME

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)=3, k=0 and l=1.

DATA

1, 3, 7, 24, 91, 376, 1635, 7377, 34197, 161876, 779125, 3801307, 18757219, 93444662, 469349303, 2374206202, 12084696935, 61848753886, 318082531211, 1643009103729, 8520055528453, 44338931718570, 231488012768833

OFFSET

0,2

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).

EXAMPLE

a(2)=2*3+1=7. a(3)=2*1*7+9+1=24. a(4)=2*1*24+2*3*7+1=91.

MAPLE

l:=1: : k := 0 : m:=3: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): od;

CROSSREFS

Cf. A176604, A176605.

KEYWORD

easy,nonn

AUTHOR

Richard Choulet (richardchoulet(AT)yahoo.fr), Apr 21 2010

STATUS

approved