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A045720 3-fold convolution of A001700(n), n >= 0. 6
1, 9, 57, 312, 1578, 7599, 35401, 161052, 719790, 3173090, 13836426, 59803104, 256596276, 1094249019, 4642178601, 19605872724, 82483419846, 345839048094, 1445715336366, 6027524015664, 25070662980876, 104056307673654 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Total number of 132 (or 213) patterns in the set of all 123-avoiding permutations of length (n+3). - Cheyne Homberger, Mar 16 2012

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..1500

José Agapito, Ângela Mestre, Maria M. Torres, and Pasquale Petrullo, On One-Parameter Catalan Arrays, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.1.

A. Ayyer, Towards a Human Proof of Gessel's Conjecture, JIS 12 (2009) 09.4.2

C. Homberger, Expected patterns in permutation classes, Electronic Journal of Combinatorics, 19(3) (2012), P43.

Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.

D. R. Snow, Spreadsheets, Power Series, Generating Functions and Integers, The College Maths. J. 20 (1989) 149.

FORMULA

a(n) = (n+5)*binomial(2*(n+3), n+3)/4 - 3*2^(2*n+3);

G.f.: (c(x)/sqrt(1-4*x))^3, where c(x) = g.f. for Catalan numbers A000108;

recursion: a(n)=(2*(2*n+7)/(n+3))*a(n-1)+(3/(n+3))*A008549(n+1), a(0)=1.

MATHEMATICA

Table[(n+5)*Binomial[2*(n+3), n+3]/4-3*2^(2n+3), {n, 0, 21}] (* Indranil Ghosh, Feb 18 2017 *)

PROG

(Python)

import math

def C(n, r):

....f=math.factorial

....return f(n)/f(r)/f(n-r)

def A045720(n):

....return (n+5)*C(2*(n+3), n+3)/4-3*2**(2*n+3) # Indranil Ghosh, Feb 18 2017

(PARI) x='x+O('x^30); Vec((((1-4*x)^(-1/2)-1)/(2*x))^3) \\ Altug Alkan, Sep 04 2018

CROSSREFS

Cf. A000108, A001700, A008549.

Sequence in context: A064838 A027210 A192054 * A014916 A045635 A026896

Adjacent sequences:  A045717 A045718 A045719 * A045721 A045722 A045723

KEYWORD

easy,nonn

AUTHOR

Wolfdieter Lang

STATUS

approved

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Last modified January 22 09:44 EST 2019. Contains 319363 sequences. (Running on oeis4.)