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A001870
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Expansion of (1-x)/(1-3*x+x^2)^2.
(Formerly M3886 N1595)
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15
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1, 5, 19, 65, 210, 654, 1985, 5911, 17345, 50305, 144516, 411900, 1166209, 3283145, 9197455, 25655489, 71293590, 197452746, 545222465, 1501460635, 4124739581, 11306252545, 30928921224, 84451726200, 230204999425
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(n)= ((n+1)*F(2*n+3)+(2*n+3)*F(2*(n+1)))/5 with F(n)=A000045(n) (Fibonacci numbers). One half of odd indexed A001629(n), n >= 2, (Fibonacci convolution).
Convolution of F(2n+1) (A001519) and F(2n+2) (A001906(n+1)) - Graeme McRae (g_m(AT)mcraefamily.com), Jun 07 2006
Number of reentrant corners along the lower contours of all directed column-convex polyominoes of area n+3 (a reentrant corner along the lower contour is a vertical step that is followed by a horizontal step). a(n)=Sum(k*A121466(n+3,k), k=0..ceil((n+1)/2)). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 02 2006
Contribution from Wolfdieter Lang, Jan 02 2012 (Start)
a(n)=A024458(2*n),n>=1 (bisection, even arguments).
a(n) is also the odd part of the bisection of the half-convolution of the sequence A000045(n+1), n>=0, with itself. See a comment on A201204 for the defintion of the half-convolution of a sequence with itself. There one also finds the rule for the o.g.f. which in this case is Chato(x)/2 with the o.g.f. Chato(x)=2*(1-x)/(1-3*x+x^2)^2 of A001629(2*n+3), n>=0.
(End)
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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.
E. Deutsch and H. Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, Theoretical Comp. Science, 307, 2003, 319-325.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Pieter Moree, Convoluted Convolved Fibonacci Numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2.
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index to sequences with linear recurrences with constant coefficients, signature (6,-11,6,-1).
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FORMULA
| a(n)=sum(k*binom(n+k+1, 2k), k=1..n+1) - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 11 2003
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MAPLE
| A001870:=-(-1+z)/(z**2-3*z+1)**2; [S. Plouffe in his 1992 dissertation.]
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CROSSREFS
| a(n)= A060921(n+1, 1)/2.
Partial sums of A030267. First differences of A001871.
Cf. A121466.
Sequence in context: A003296 A053545 A049612 * A025568 A001047 A099448
Adjacent sequences: A001867 A001868 A001869 * A001871 A001872 A001873
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Christian G. Bower (bowerc(AT)usa.net).
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