A331576 - OEIS

%I #32 Sep 08 2022 08:46:25

%S 0,0,0,0,0,0,0,0,0,0,1,6,36,126,441,1176,3136,7056,15876,31752,63504,

%T 116424,213444,365904,627264,1019304,1656369,2576574,4008004,6012006,

%U 9018009,13117104,19079424,27029184,38291344,53018784,73410624,99628704,135210384,180280512,240374016,315490896

%N a(n) is the number of subsets of {1..n} that contain 5 even and 5 odd numbers.

%C In general, the number of subsets of {1..n} that contain k even and k odd numbers is given by binomial(n/2, k)^2 for n even and binomial((n-1)/2, k)*binomial((n+1)/2, k) for n odd.

%H Colin Barker, <a href="/A331576/b331576.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_20">Index entries for linear recurrences with constant coefficients</a>, signature (2,8,-18,-27,72,48,-168,-42,252,0,-252,42,168,-48,-72,27,18,-8,-2,1).

%F a(n) = binomial(n/2, 5)^2, for n even;

%F a(n) = binomial((n-1)/2, 5)*binomial((n+1)/2,5), for n odd.

%F From _Colin Barker_, Jan 21 2020: (Start)

%F 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).

%F 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.

%F (End)

%F 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

%e 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}.

%p a:= n-> ((b, q)-> b(q, 5)*b(n-q, 5))(binomial, iquo(n, 2)):

%p seq(a(n), n=0..50);  # _Alois P. Heinz_, Jan 30 2020

%t 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 *)

%o (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

%o (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

%Y Cf. A028723 (k=2), A331574 (k=3), A331575 (k=4). See comment.

%K nonn,easy

%O 0,12

%A _Enrique Navarrete_, Jan 20 2020