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A038731
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Number of columns in all directed column-convex polyominoes of area n+1.
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6
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1, 3, 10, 32, 99, 299, 887, 2595, 7508, 21526, 61251, 173173, 486925, 1362627, 3797374, 10543724, 29180067, 80521055, 221610563, 608468451, 1667040776, 4558234018, 12441155715, 33900136297, 92230468249, 250570010499, 679844574322, 1842280003640
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Apply Riordan array (1/(1-x), x/(1-x)^2) to n+1. [From Paul Barry (pbarry(AT)wit.ie), Oct 13 2009]
Binomial transform of (A001629 shifted left twice) [R. J. Mathar, (mathar(AT)strw.leidenuniv.nl), Feb 06 2010]
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REFERENCES
| E. Barcucci, R. Pinzani and R. Sprugnoli, Directed column-convex polyominoes by recurrence relations, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..200
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FORMULA
| a_n = ((2n+1)/5)F(2n+2)-((n-4)/5)F(2n+1), where the F(n)'s are the Fibonacci numbers, F(0)=0, F(1)=1
a(n)=sum(k*binom(n+k-1, 2k-2), k=1..n+1) - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 11 2003
Contribution from Paul Barry (pbarry(AT)wit.ie), Oct 13 2009: (Start)
G.f.: (1-x)^3/(1-3x+x^2)^2.
a(n)=sum{k=0..n, C(n+k,2k)(k+1)}. (End)
a(n)= 6*a(n-1) -11*a(n-2) +6*a(n-3) -a(n-4). [R. J. Mathar, (mathar(AT)strw.leidenuniv.nl), Feb 06 2010]
a(n)=sum{k=0..n, (F(2k)+0^k)*F(2n-2k+1)}. [From Paul Barry (pbarry(AT)wit.ie), Jun 23 2010]
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MATHEMATICA
| Table[Sum[Binomial[n, k]*CoefficientList[Series[1/(1 - x - x^2)^2, {x, 0, k}], x][[-1]], {k, 0, n}], {n, 0, 27}] (* Arkadiusz Wesolowski, Feb 03 2012 *)
LinearRecurrence[{6, -11, 6, -1}, {1, 3, 10, 32}, 30] (* Vincenzo Librandi, Feb 04 2012 *)
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PROG
| (MAGMA) I:=[1, 3, 10, 32]; [n le 4 select I[n] else 6*Self(n-1)-11*Self(n-2)+6*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Feb 04 2012
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CROSSREFS
| Row-sums of array T as in A038730.
First differences of A030267.
Sequence in context: A080406 A036682 A104270 * A053581 A092822 A017935
Adjacent sequences: A038728 A038729 A038730 * A038732 A038733 A038734
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KEYWORD
| nonn,changed
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu), May 02 2000
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EXTENSIONS
| Entry improved by comments from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 14 2001
Corrected typo in the denominator of the g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 06 2010
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