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A262977
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a(n) = binomial(4*n-1,n).
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24
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1, 3, 21, 165, 1365, 11628, 100947, 888030, 7888725, 70607460, 635745396, 5752004349, 52251400851, 476260169700, 4353548972850, 39895566894540, 366395202809685, 3371363686069236, 31074067324187580, 286845713747883300, 2651487106659130740, 24539426037817994160
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OFFSET
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0,2
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COMMENTS
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Also the number of integer compositions of 4n with alternating sum 2n, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. These compositions are ranked by A348614. The a(12) = 21 compositions are:
(6,2) (1,2,5) (1,1,5,1) (1,1,1,1,4)
(2,2,4) (2,1,4,1) (1,1,2,1,3)
(3,2,3) (3,1,3,1) (1,1,3,1,2)
(4,2,2) (4,1,2,1) (1,1,4,1,1)
(5,2,1) (5,1,1,1) (2,1,1,1,3)
(2,1,2,1,2)
(2,1,3,1,1)
(3,1,1,1,2)
(3,1,2,1,1)
(4,1,1,1,1)
The following pertain to this interpretation:
- Allowing any alternating sum gives A013777 (compositions of 4n).
- A011782 counts compositions of n.
- A034871 counts compositions of 2n with alternating sum 2k.
- A097805 counts compositions by alternating (or reverse-alternating) sum.
- A103919 counts partitions by sum and alternating sum (reverse: A344612).
- A345197 counts compositions by length and alternating sum.
(End)
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LINKS
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FORMULA
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G.f.: A(x)=x*B'(x)/B(x), where B(x) if g.f. of A006632.
a(n) = Sum_{k=0..n}(binomial(n-1,n-k)*binomial(3*n,k)).
The o.g.f. equals f(x)/g(x), where f(x) is the o.g.f. for A005810 and g(x) is the o.g.f. for A002293. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(4*n + k,n). Cf. A005810 (k = 0), A052203 (k = 1), A257633 (k = 2), A224274 (k = 3) and A004331 (k = 4). (End)
a(n) = (-1)^n * binomial(-3*n, n).
a(n) = hypergeom([1 - 3*n, -n], [1], 1).
The g.f. A(x) satisfies A(x/(1 + x)^4) = 1/(1 - 3*x). (End)
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MATHEMATICA
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PROG
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(Maxima)
B(x):=sum(binomial(4*n-1, n-1)*3/(4*n-1)*x^n, n, 1, 30);
taylor(x*diff(B(x), x, 1)/B(x), x, 0, 20);
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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