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Search: id:A071879
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| A071879 |
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Coefficients of power series solution to g(x) = 1 + x*g(x) + (x*g(x))^3. The first-order differences of these coefficients of g(x), where: g(x) = 1 + 1x + 1x^2 + 2x^3 + 5x^4 + 11x^5 + 24x^6 + ... + a(n)*x^n + ..., forms the coefficients of the third power of g(x), where: g(x)^3 = 1 + 3x + 6x^2 + 13x^3 + 33x^4 + 84x^5 + 208x^6 + 522x^7 + ... |
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+0
2
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| 1, 1, 1, 2, 5, 11, 24, 57, 141, 349, 871, 2212, 5688, 14730, 38403, 100829, 266333, 706997, 1885165, 5047522, 13565203, 36578497, 98934826, 268342933, 729709432, 1989021256, 5433518806, 14873285506, 40790118487, 112064912455
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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Number of ordered trees with n edges and having nonleaf nodes of outdegrees 1 or 3. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 03 2002. [Comment corrected by Christian Bower, Sep 25 2007.]
Sequence is a Motzkin-like sequence. The Motzkin sequence A001006 counts ordered trees with n edges and having nodes of outdegree 0, 1, or 2 [g.f. f(x) defined by f = 1+x*f+(x*f)^2]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 30 2007
G.f. (offset 1) is series reversion of x^2/(x+x^2+x^4).
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FORMULA
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a(n)=sum(binomial(n+1, 1+2i)*binomial(n-2i, i), i=0..floor(n/3))/(n+1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 03 2002
a(n)=sum{k=0..floor(n/3), C(n,3k)C(3k,k)/(2k+1)}; - Paul Barry (pbarry(AT)wit.ie), Sep 07 2006
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MAPLE
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a := n->sum(binomial(n+1, 1+2*i)*binomial(n-2*i, i), i=0..floor(n/3))/(n+1): seq(a(n), n=0..29);
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MATHEMATICA
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a[n_] := Sum[Binomial[n+1, 1+2i]*Binomial[n-2i, i], {i, 0, Floor[n/3]}]/(n+1);
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PROGRAM
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(PARI) a(n)=if(n<0, 0, polcoeff(serreverse(x^2/(x+x^2+x^4+x^2*O(x^n))), n+1))
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CROSSREFS
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Sequence in context: A110513 A018115 A018007 this_sequence A134527 A124379 A084978
Adjacent sequences: A071876 A071877 A071878 this_sequence A071880 A071881 A071882
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Jun 10 2002
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