This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A331575 #28 Jul 20 2025 19:12:14
%S A331575 0,0,0,0,0,0,0,0,1,5,25,75,225,525,1225,2450,4900,8820,15876,26460,
%T A331575 44100,69300,108900,163350,245025,353925,511225,715715,1002001,
%U A331575 1366365,1863225,2484300,3312400,4331600,5664400,7282800,9363600,11860560,15023376,18779220,23474025,28997325
%N A331575 a(n) is the number of subsets of {1..n} that contain 4 even and 4 odd numbers.
%H A331575 Colin Barker, <a href="/A331575/b331575.txt">Table of n, a(n) for n = 0..1000</a>
%H A331575 <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (2,6,-14,-14,42,14,-70,0,70,-14,-42,14,14,-6,-2,1).
%F A331575 a(n) = binomial(n/2,4)^2, n even;
%F A331575 a(n) = binomial((n-1)/2,4)*binomial((n+1)/2,4), n odd.
%F A331575 From _Colin Barker_, Jan 21 2020: (Start)
%F A331575 G.f.: x^8*(1 + 3*x + 9*x^2 + 9*x^3 + 9*x^4 + 3*x^5 + x^6) / ((1 - x)^9*(1 + x)^7).
%F A331575 a(n) = 2*a(n-1) + 6*a(n-2) - 14*a(n-3) - 14*a(n-4) + 42*a(n-5) + 14*a(n-6) - 70*a(n-7) + 70*a(n-9) - 14*a(n-10) - 42*a(n-11) + 14*a(n-12) + 14*a(n-13) - 6*a(n-14) - 2*a(n-15) + a(n-16) for n>15.
%F A331575 (End)
%F A331575 E.g.f.: (cosh(x)-sinh(x))*(1575+1350*x+630*x^2+204*x^3+54*x^4+12*x^5+4*x^6+(-1575+1800*x-1080*x^2+456*x^3-156*x^4+48*x^5-16*x^6+8*x^7+2*x^8)*(cosh(2*x)+sinh(2*x)))/294912. - _Stefano Spezia_, Jan 27 2020
%e A331575 a(9)=5 and the 5 subsets are {1,2,3,4,5,6,7,8}, {1,2,3,4,5,6,8,9}, {1,2,3,4,6,7,8,9}, {1,2,4,5,6,7,8,9}, {2,3,4,5,6,7,8,9}.
%p A331575 a:= n-> ((b, q)-> b(q, 4)*b(n-q, 4))(binomial, iquo(n, 2)):
%p A331575 seq(a(n), n=0..50); # _Alois P. Heinz_, Jan 30 2020
%t A331575 a[n_] := If[OddQ[n], Binomial[(n - 1)/2, 4]*Binomial[(n + 1)/2, 4], Binomial[n/2, 4]^2]; Array[a, 42, 0] (* _Amiram Eldar_, Jan 21 2020 *)
%t A331575 LinearRecurrence[{2,6,-14,-14,42,14,-70,0,70,-14,-42,14,14,-6,-2,1},{0,0,0,0,0,0,0,0,1,5,25,75,225,525,1225,2450},50] (* _Harvey P. Dale_, Jul 20 2025 *)
%o A331575 (PARI) a(n) = if (n%2, binomial((n-1)/2,4)*binomial((n+1)/2,4), binomial(n/2,4)^2); \\ _Michel Marcus_, Jan 21 2020
%o A331575 (PARI) concat([0,0,0,0,0,0,0,0], Vec(x^8*(1 + 3*x + 9*x^2 + 9*x^3 + 9*x^4 + 3*x^5 + x^6) / ((1 - x)^9*(1 + x)^7) + O(x^40))) \\ _Colin Barker_, Jan 21 2020
%o A331575 (Magma) [IsEven(n) select Binomial((n div 2),4)^2 else Binomial((n-1) div 2,4)*Binomial((n+1) div 2,4): n in [0..41]]; // _Marius A. Burtea_, Jan 21 2020
%Y A331575 Cf. A028723, A331574.
%Y A331575 Cf. A288876 (even bisection, shifted).
%K A331575 nonn,easy
%O A331575 0,10
%A A331575 _Enrique Navarrete_, Jan 20 2020