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A072863 Binomial transform of n^2/2 - n/2 + 1. 5
1, 3, 9, 26, 72, 192, 496, 1248, 3072, 7424, 17664, 41472, 96256, 221184, 503808, 1138688, 2555904, 5701632, 12648448, 27918336, 61341696, 134217728, 292552704, 635437056, 1375731712, 2969567232, 6392119296, 13723762688 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Number of 123-avoiding ternary words of length n-1.

Equals row sums of triangle A144333. - Gary W. Adamson, Sep 18 2008

LINKS

Table of n, a(n) for n=1..28.

P. Braendeen and T. Mansour, Finite automata and pattern avoidance in words

Index entries for linear recurrences with constant coefficients, signature (6,-12,8)

FORMULA

f[n_, 1] := n^2/2 - n/2 + 1; f[n_, m_] := f[n, m] = f[n, m - 1] + f[n + 1, m - 1].

G.f. : (1-3x+3x^2)/(1-2x)^3; a(n)=2^(n-3)(n^2+3n+8). - Paul Barry, Jul 22 2004

E.g.f.: e^(2x)*(1+x+x^2/2); a(n)=sum{k=0..2, C(n,k)*2^(n-k)} [offset 0]; - Paul Barry, Mar 27 2007

Row sums of triangle A134247. Also, binomial transform of A000124: (1, 2, 4, 7, 11, 16, 22, 29,...) and double binomial transform of (1, 1, 1, 0, 0, 0,...). - Gary W. Adamson, Oct 15 2007

MAPLE

with(combstruct); gramm_Alkyl:=Alkyl=Prod(Carbon, Set(Alkyl, card<=1)), Carbon=Atom: specs_Alkyl:=[Alkyl, {gramm_Alkyl}, unlabeled]: gramm_S1_Alkyl:=S1_Alkyl[X]=Union(Prod(Carbon, S1_Alkyl[X], Set(Alkyl, card<=1)), Prod(Prod(Carbon, X), Set(Alkyl, card<=1))), X=Epsilon: specs_S1_Alkyl:=[S1_Alkyl[X], {gramm_S1_Alkyl, gramm_Alkyl}, unlabeled]: gramm_S2b_Alkyl:=S2_Alkyl[X, X]=Union(Prod(Carbon, S2_Alkyl[X, X], Set(Alkyl, card<=1)), Prod(Carbon, Union(Prod(S1_Alkyl[X], S1_Alkyl[X]), Prod(S1_Alkyl[X], X), Prod(X, X)), Set(Alkyl, card<=1))): specs_S2b_Alkyl:=[S2_Alkyl[X, X], {gramm_S2b_Alkyl, gramm_S1_Alkyl, gramm_Alkyl}, unlabeled]: seq(count(specs_S2b_Alkyl, size=i), i=1..28); # - Zerinvary Lajos, Apr 15 2009

MATHEMATICA

Table[Sum[Binomial[m-1, k](#^2/2 -#/2 +1 &)[k+1], {k, 0, m}], {m, 36}]

PROG

(PARI) a(n)=2^(n-3)*(n^2+3*n+8) \\ Charles R Greathouse IV, Oct 07 2015

CROSSREFS

Cf. A134247, A000124, A144333.

Sequence in context: A048470 A138237 A121286 * A054963 A118046 A057153

Adjacent sequences:  A072860 A072861 A072862 * A072864 A072865 A072866

KEYWORD

nonn,easy

AUTHOR

Michael A. Childers (childers_moof(AT)yahoo.com), Jul 27 2002

EXTENSIONS

Corrected and extended by Wouter Meeussen, Jul 30 2002

STATUS

approved

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Last modified May 27 22:57 EDT 2017. Contains 287210 sequences.