A331576 a(n) is the number of subsets of {1..n} that contain 5 even and 5 odd numbers.
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 36, 126, 441, 1176, 3136, 7056, 15876, 31752, 63504, 116424, 213444, 365904, 627264, 1019304, 1656369, 2576574, 4008004, 6012006, 9018009, 13117104, 19079424, 27029184, 38291344, 53018784, 73410624, 99628704, 135210384, 180280512, 240374016, 315490896
Offset: 0
Examples
a(11)=6 and the 6 subsets are {1,2,3,4,5,6,7,8,9,10}, {1,2,3,4,5,6,7,8,10,11}, {1,2,3,4,5,6,8,9,10,11}, {1,2,3,4,6,7,8,9,10,11}, {1,2,4,5,6,7,8,9,10,11}, {2,3,4,5,6,7,8,9,10,11}.
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (2,8,-18,-27,72,48,-168,-42,252,0,-252,42,168,-48,-72,27,18,-8,-2,1).
Programs
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Magma
[IsOdd(n) select Binomial((n-1) div 2,5)*Binomial((n+1) div 2,5) else Binomial(n div 2,5)^2: n in [0..41]]; // Marius A. Burtea, Jan 21 2020
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Maple
a:= n-> ((b, q)-> b(q, 5)*b(n-q, 5))(binomial, iquo(n, 2)): seq(a(n), n=0..50); # Alois P. Heinz, Jan 30 2020
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Mathematica
a[n_] := If[OddQ[n], Binomial[(n - 1)/2, 5]*Binomial[(n + 1)/2, 5], Binomial[n/2, 5]^2]; Array[a, 42, 0] (* Amiram Eldar, Jan 21 2020 *)
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PARI
concat([0,0,0,0,0,0,0,0,0,0], Vec(x^10*(1 + 4*x + 16*x^2 + 24*x^3 + 36*x^4 + 24*x^5 + 16*x^6 + 4*x^7 + x^8) / ((1 - x)^11*(1 + x)^9) + O(x^40))) \\ Colin Barker, Jan 21 2020
Formula
a(n) = binomial(n/2, 5)^2, for n even;
a(n) = binomial((n-1)/2, 5)*binomial((n+1)/2,5), for n odd.
From Colin Barker, Jan 21 2020: (Start)
G.f.: x^10*(1 + 4*x + 16*x^2 + 24*x^3 + 36*x^4 + 24*x^5 + 16*x^6 + 4*x^7 + x^8) / ((1 - x)^11*(1 + x)^9).
a(n) = 2*a(n-1) + 8*a(n-2) - 18*a(n-3) - 27*a(n-4) + 72*a(n-5) + 48*a(n-6) - 168*a(n-7) - 42*a(n-8) + 252*a(n-9) - 252*a(n-11) + 42*a(n-12) + 168*a(n-13) - 48*a(n-14) - 72*a(n-15) + 27*a(n-16) + 18*a(n-17) - 8*a(n-18) - 2*a(n-19) + a(n-20) for n>19.
(End)
E.g.f.: (cosh(x) - sinh(x))*(99225 + 88200*x + 40950*x^2 + 13050*x^3 + 3225*x^4 + 660*x^5 + 120*x^6 + 20*x^7 + 5*x^8 + (-99225 + 110250*x - 63000*x^2 + 24750*x^3 - 7575*x^4 + 1950*x^5 - 450*x^6 + 100*x^7 - 25*x^8 + 10*x^9 + 2*x^10)*(cosh(2*x) + sinh(2*x)))/29491200. - Stefano Spezia, Jan 27 2020
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