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A052544
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Expansion of (1-x)^2/(1-4x+3x^2-x^3).
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5
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1, 2, 6, 19, 60, 189, 595, 1873, 5896, 18560, 58425, 183916, 578949, 1822473, 5736961, 18059374, 56849086, 178955183, 563332848, 1773314929, 5582216355, 17572253481, 55315679788, 174128175064, 548137914373, 1725482812088
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Equals INVERT transform of (1, 1, 3, 8, 21, 55, 144,...). [From Gary W. Adamson, May 01 2009]
The Ze2 sums, see A180662, of triangle A065941 equal the terms (doubled) of this sequence. [Johannes W. Meijer, Aug 16 2011]
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..500
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 480
H. Stephan, Rekursive Folgen im Pascalschen Dreieck
Index to sequences with linear recurrences with constant coefficients, signature (4,-3,1).
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FORMULA
| G.f.: -(-1+x)^2/(-1+4*x-3*x^2+x^3)
a(n) = 4*a(n-1) - 3*a(n-2) + a(n-3).
Sum(-1/31*(-4-7*_alpha+2*_alpha^2)*_alpha^(-1-n), _alpha=RootOf(-1+4*_Z-3*_Z^2+_Z^3))
a(n) = Sum(binomial(n+2*k, 3*k), {k=0...n}) - Richard Ollerton (r.ollerton(AT)uws.edu.au), May 12 2004
G.f.: 1 / (1 - x - x / (1 - x)^2). - Michael Somos, Jan 12 2012
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EXAMPLE
| 1 + 2*x + 6*x^2 + 19*x^3 + 60*x^4 + 189*x^5 + 595*x^6 + 1873*x^7 + ...
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MAPLE
| spec := [S, {S=Sequence(Union(Z, Prod(Z, Sequence(Z), Sequence(Z))))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..25);
A052544 := proc(n): add(binomial(n+2*k, 3*k), k=0...n) end: seq(A052544(n), n=0..25); # [Johannes W. Meijer, Aug 16 2011]
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MATHEMATICA
| LinearRecurrence[{4, -3, 1}, {1, 2, 6}, 30] (* From Harvey P. Dale, Jul 13 2011 *)
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PROG
| {a(n) = sum( k=0, n, binomial( n + 2*k, 3*k))} /* Michael Somos, Jan 12 2012 */
(MAGMA) I:=[1, 2, 6]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jan 12 2012
(PARI) Vec((1-x)^2/(1-4*x+3*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Jan 12 2012
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CROSSREFS
| Cf. A124820 (partial sums).
Sequence in context: A183188 A118364 * A204200 A001169 A187276 A022041
Adjacent sequences: A052541 A052542 A052543 * A052545 A052546 A052547
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KEYWORD
| easy,nonn
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AUTHOR
| encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jun 06 2000
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