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 A333319 a(n) is the number of subsets of {1..n} that contain exactly 3 odd and 1 even numbers. 1
 0, 0, 0, 0, 0, 2, 3, 12, 16, 40, 50, 100, 120, 210, 245, 392, 448, 672, 756, 1080, 1200, 1650, 1815, 2420, 2640, 3432, 3718, 4732, 5096, 6370, 6825, 8400, 8960, 10880, 11560, 13872, 14688, 17442, 18411, 21660, 22800, 26600, 27930, 32340, 33880, 38962, 40733, 46552 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS The general formula for the number of subsets of {1..n} that contain exactly k odd and j even numbers is binomial(ceiling(n/2), k) * binomial(floor(n/2), j). LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1). FORMULA a(n) = binomial(ceiling(n/2),3) * binomial(floor(n/2), 1). From Colin Barker, Mar 15 2020: (Start) G.f.: x^5*(2 + x + x^2) / ((1 - x)^5*(1 + x)^4). a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n>8. (End) EXAMPLE a(7)=12 and the 12 subsets are {1,2,3,5}, {1,2,3,7}, {1,2,5,7}, {1,3,4,5}, {1,3,4,7}, {1,3,5,6}, {1,3,6,7}, {1,4,5,7}, {1,5,6,7}, {2,3,5,7}, {3,4,5,7}, {3,5,6,7}. MATHEMATICA Array[Binomial[Ceiling[#], 3] Binomial[Floor[#], 1] &[#/2] &, 48, 0] (* Michael De Vlieger, Mar 14 2020 *) PROG (PARI) concat([0, 0, 0, 0, 0], Vec(x^5*(2 + x + x^2) / ((1 - x)^5*(1 + x)^4) + O(x^40))) \\ Colin Barker, Mar 15 2020 CROSSREFS Cf. A330299. Sequence in context: A260908 A123761 A181121 * A047163 A046486 A073452 Adjacent sequences:  A333316 A333317 A333318 * A333320 A333321 A333322 KEYWORD nonn,easy AUTHOR Enrique Navarrete, Mar 14 2020 STATUS approved

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Last modified January 21 06:14 EST 2021. Contains 340333 sequences. (Running on oeis4.)