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A331419 a(n) is the number of subsets of {1..n} that contain 4 odd numbers. 2
0, 0, 0, 0, 0, 0, 8, 16, 80, 160, 480, 960, 2240, 4480, 8960, 17920, 32256, 64512, 107520, 215040, 337920, 675840, 1013760, 2027520, 2928640, 5857280, 8200192, 16400384, 22364160, 44728320, 59637760, 119275520, 155975680, 311951360, 401080320, 802160640, 1016070144, 2032140288 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

2*a(n-1) for n > 1 is the number of subsets of {1..n} that contain 4 even numbers.  For example, for n=8, 2*a(7)=16 and the 16 subsets are {2,4,6,8}, {1,2,4,6,8}, {2,3,4,6,8}, {2,4,5,6,8}, {2,4,6,7,8}, {1,2,3,4,6,8}, {1,2,4,5,6,8}, {1,2,4,6,7,8}, {2,3,4,5,6,8}, {2,3,4,6,7,8}, {2,4,5,6,7,8}, {1,2,3,4,5,6,8}, {1,2,3,4,6,7,8}, {1,2,4,5,6,7,8}, {2,3,4,5,6,7,8}, {1,2,3,4,5,6,7,8}.

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0,10,0,-40,0,80,0,-80,0,32).

FORMULA

a(n) = binomial((n+1)/2, 4) * 2^((n-1)/2), n odd;

a(n) = binomial((n/2), 4) * 2^(n/2), n even.

From Colin Barker, Jan 18 2020: (Start)

G.f.: 8*x^7*(1 + 2*x) / (1 - 2*x^2)^5.

a(n) = 10*a(n-2) - 40*a(n-4) + 80*a(n-6) - 80*a(n-8) + 32*a(n-10) for n>10.

(End)

EXAMPLE

a(7)=8 and the 8 subsets are {1,3,5,7}, {1,2,3,5,7}, {1,3,4,5,7}, {1,3,5,6,7}, {1,2,3,4,5,7}, {1,2,3,5,6,7}, {1,3,4,5,6,7}, {1,2,3,4,5,6,7}.

MATHEMATICA

a[n_] := If[OddQ[n], Binomial[(n + 1)/2, 4]*2^((n - 1)/2), Binomial[n/2, 4]*2^(n/2)]; Array[a, 38] (* Amiram Eldar, Jan 17 2020 *)

PROG

(MAGMA) [IsOdd(n) select Binomial((n+1) div 2, 4)*2^((n-1) div 2) else Binomial((n div 2), 4)*2^(n div 2): n in [1..38]]; // Marius A. Burtea, Jan 17 2020

(PARI) concat([0, 0, 0, 0, 0, 0], Vec(8*x^7*(1 + 2*x) / (1 - 2*x^2)^5 + O(x^40))) \\ Colin Barker, Jan 18 2020

CROSSREFS

Cf. A330592, A331408.

Sequence in context: A218066 A157164 A131539 * A271080 A117868 A291001

Adjacent sequences:  A331416 A331417 A331418 * A331420 A331421 A331422

KEYWORD

nonn,easy

AUTHOR

Enrique Navarrete, Jan 16 2020

STATUS

approved

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Last modified January 27 15:31 EST 2021. Contains 340467 sequences. (Running on oeis4.)