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A352027
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a(n) = binomial(2*n-1,n) - n*(n-1) - 1.
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4
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0, 0, 3, 22, 105, 431, 1673, 6378, 24237, 92287, 352605, 1351945, 5200143, 20058117, 77558549, 300539954, 1166802837, 4537567343, 17672631557, 68923264029, 269128936799, 1052049481397, 4116715363293, 16123801840997, 63205303218275, 247959266473401, 973469712823353
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OFFSET
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1,3
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COMMENTS
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a(n) is the number of ways to place n indistinguishable balls into n distinguishable boxes with at least 2 boxes remaining empty.
a(n) is also the number of weak compositions of n into n parts in which at least two parts are zero.
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LINKS
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FORMULA
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G.f.: 2*x/(4*x - 1 + sqrt(1 - 4*x)) - (1 - 2*x + 3*x^2)/(1 - x)^3. - Stefano Spezia, Mar 01 2022
D-finite with recurrence n*a(n) +2*(-3*n+2)*a(n-1) +(9*n-14)*a(n-2) +2*(-2*n+5)*a(n-3) +2*(-9*n+22)=0. - R. J. Mathar, Jan 25 2023
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EXAMPLE
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a(4)=22 since 4 can be written as 3+1+0+0, 0+3+0+1, etc. (12 such compositions); 2+2+0+0 (6 such compositions); 4+0+0+0 (4 such compositions).
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MATHEMATICA
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Table[Binomial[2n-1, n]-n(n-1)-1, {n, 40}] (* Harvey P. Dale, Dec 03 2022 *)
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PROG
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(PARI) a(n) = binomial(2*n-1, n) - n*(n-1) - 1; \\ Michel Marcus, Apr 12 2022
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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