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 A282876 Expansion of ((1 + 4*x + 8*x^2)^(3/2) - (1 + 6*x + 18*x^2 + 20*x^3)) / (2*x^4) in powers of x. 0
 3, -6, 10, -12, 3, 34, -114, 204, -114, -636, 2676, -5528, 3939, 17778, -83994, 186972, -150438, -609524, 3091020, -7204008, 6237902, 23649204, -125807412, 302476536, -275144388, -996903096, 5489607272, -13498689840, 12721569699, 44596212754, -252074322858 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 LINKS Table of n, a(n) for n=0..30. Ewan Delanoy, Divisibility property for sequence a(n+2) = -2(n-1)(n+3)a(n) - (2n+3)a(n+1), Mathematics Stack Exchange question 2728009, Apr 08 2018. FORMULA 0 = (8*n + 8)*a(n) + (4*n + 14)*a(n+1) + (n + 6)*a(n+2) for all n in Z if a(-1)=10, a(-2)=9, a(-3)=3, a(-4)=1/2, and also 0 = a(n)*(+64*a(n+1) +112*a(n+2) +48*a(n+3)) +a(n+1)*(-48*a(n+1) -16*a(n+2) +14*a(n+3)) +a(n+2)*(-6*a(n+2) +a(n+3)) for all n in Z. D-finite with recurrence (n+4)*a(n) +2*(2*n+3)*a(n-1) +8*(n-1)*a(n-2)=0. - R. J. Mathar, Sep 24 2021 EXAMPLE G.f. = 3 - 6*x + 10*x^2 - 12*x^3 + 3*x^4 + 34*x^5 - 114*x^6 + 204*x^7 + ... MATHEMATICA a[ n_] := If[ n < 1, 3 Boole[n==0], Sum[ (-1)^k Binomial[k, 2 k - n - 4] (2 k - 5)! / (2^(k - 3) k! (k - 3)!), {k, 3, n + 4}] 24 2^n]; PROG (PARI) {a(n) = if( n<1, 3*(n==0), sum(k=3, n+4, (-1)^k * binomial(k, 2*k-n-4) * (2*k-5)! / (2^(k-3) * k! * (k-3)!)) * 24 * 2^n)}; CROSSREFS Sequence in context: A351827 A351828 A158975 * A363775 A261662 A345915 Adjacent sequences: A282873 A282874 A282875 * A282877 A282878 A282879 KEYWORD sign AUTHOR Michael Somos, Oct 26 2018 STATUS approved

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Last modified June 13 00:18 EDT 2024. Contains 373362 sequences. (Running on oeis4.)