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A176605 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=0 and l=1. 3
1, 1, 3, 8, 23, 72, 239, 825, 2929, 10624, 39193, 146587, 554535, 2118042, 8156595, 31635298, 123462515, 484483902, 1910465543, 7566438417, 30084771297, 120044573286, 480550302501, 1929362833770, 7767140703837, 31346346634338 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..25.

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) +(9*n-13)*a(n-2) +8*(-n+3)*a(n-3) +4*(n-4)*a(n-4)=0. - R. J. Mathar, Feb 29 2016

a(n) = Sum_{k=0..n}((C(k)*Sum_{j=0..(n-k)/2}(binomial(k+1,j)*binomial(n-k-j-1,n-k-2*j)))), where C(n) is Catalan numbers (A000108). - Vladimir Kruchinin, Apr 15 2016

EXAMPLE

a(2)=(1*1+0)+(1*1+0)+1=3. a(3)=1*3+1^1+3*1+1=8. a(4)=2*1*8+2*1*3+1=23.

MAPLE

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

MATHEMATICA

Table[Sum[(Binomial[2 k, k] Sum[Binomial[k + 1, j] Binomial[n - k - j - 1, n - k - 2 j], {j, 0, (n - k)/2}])/(k + 1), {k, 0, n}], {n, 0, 25}] (* Michael De Vlieger, Apr 15 2016 *)

PROG

(Maxima)

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

CROSSREFS

Cf. A176604.

Sequence in context: A127385 A152880 A259441 * A080410 A230952 A148777

Adjacent sequences:  A176602 A176603 A176604 * A176606 A176607 A176608

KEYWORD

nonn

AUTHOR

Richard Choulet, Apr 21 2010

STATUS

approved

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Last modified August 1 07:45 EDT 2021. Contains 346384 sequences. (Running on oeis4.)