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A052967 Expansion of (1-x)/(1-2*x-x^2+x^4). 0
1, 1, 3, 7, 16, 38, 89, 209, 491, 1153, 2708, 6360, 14937, 35081, 82391, 193503, 454460, 1067342, 2506753, 5887345, 13826983, 32473969, 76268168, 179122960, 420687105, 988023201, 2320465339, 5449830919, 12799440072, 30060687862 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Equals INVERT transform of (1, 2, 2, 1, 1, 1,...). [From Gary W. Adamson, Apr 28 2009]

LINKS

Table of n, a(n) for n=0..29.

Shanzhen Gao, Keh-Hsun Chen, Tackling Sequences From Prudent Self-Avoiding Walks, FCS'14, The 2014 International Conference on Foundations of Computer Science.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1039

Index entries for linear recurrences with constant coefficients, signature (2,1,0,-1).

FORMULA

Recurrence: {a(1)=1, a(0)=1, a(2)=3, a(3)=7, a(n)-a(n+2)-2*a(n+3)+a(n+4)}

Sum(-1/106*(-17-22*_alpha+10*_alpha^2+8*_alpha^3)*_alpha^(-1-n), _alpha=RootOf(1-2*_Z-_Z^2+_Z^4))

a(n) = Sum_{k=0..n}(Sum_{l=0..k}(binomial(k,l)*Sum_{i=0..n-k-l}(binomial(l,i)*binomial(n-i-2*l-1,n-k-i-l)))). - Vladimir Kruchinin, Mar 16 2016

MAPLE

spec := [S, {S=Sequence(Prod(Union(Prod(Z, Z), Z, Sequence(Z)), Z))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20);

PROG

(Maxima)

a(n):=sum(sum(binomial(k, l)*sum(binomial(l, i)*binomial(n-i-2*l-1, n-k-i-l), i, 0, n-k-l), l, 0, k), k, 0, n); /* Vladimir Kruchinin, Mar 16 2016  */

(PARI) Vec((1-x)/(1-2*x-x^2+x^4) + O(x^40)) \\ Michel Marcus, Mar 16 2016

CROSSREFS

Sequence in context: A095263 A010912 A192665 * A239040 A211278 A196154

Adjacent sequences:  A052964 A052965 A052966 * A052968 A052969 A052970

KEYWORD

easy,nonn

AUTHOR

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

STATUS

approved

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Last modified March 24 21:47 EDT 2017. Contains 283997 sequences.