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 A271785 a(n) = Sum_{k=0..(n-1)/2} (n+2-k)*binomial(n-1-k,k). 2
 0, 3, 4, 9, 16, 30, 54, 97, 172, 303, 530, 922, 1596, 2751, 4724, 8085, 13796, 23478, 39858, 67517, 114140, 192603, 324454, 545714, 916536, 1537275, 2575204, 4308897, 7201912, 12025038, 20058990, 33430297, 55667596, 92622471, 153992954, 255842890 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS This is a sibling to the expansions A001629(n+1) = Sum_{k=0..(n-1)/2} (n-k) *binomial(n-1-k,k) and A226432(n+3) = Sum_{k=0..(n-1)/2} (n+1-k) *binomial(n-1-k,k). LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1). FORMULA G.f.: x*(3-2*x-2*x^2) / (1-x-x^2)^2. a(n) = 3*A001629(n+1) -2*A001629(n) -2*A001629(n-1). From Colin Barker, Apr 14 2016: (Start) a(n) = (2^(-1-n)*(-24*sqrt(5)*((1-sqrt(5))^n-(1+sqrt(5))^n)+5*((1-sqrt(5))^(1+n)+(1+sqrt(5))^(1+n))*n))/25. a(n) = 2*a(n-1)+a(n-2)-2*a(n-3)-a(n-4) for n>3. (End) E.g.f.: (1/25)*(sqrt(5)*(5*x + 24)*sinh((sqrt(5)*x)/2) + 15*x*cosh((sqrt(5)*x)/2))*exp(x/2). - Ilya Gutkovskiy, Apr 14 2016 a(n) = A006355(n+1)+A001629(n+1). - R. J. Mathar, May 20 2016 MAPLE A271785 := proc(n) add( (n+2-k)*binomial(n-1-k, k), k=0..(n-1)/2) ; end proc: MATHEMATICA LinearRecurrence[{2, 1, -2, -1}, {0, 3, 4, 9}, 40] (* Harvey P. Dale, May 05 2020 *) PROG (PARI) concat(0, Vec(x*(3-2*x-2*x^2)/(1-x-x^2)^2 + O(x^50))) \\ Colin Barker, Apr 14 2016 CROSSREFS Cf. A001629. Sequence in context: A336450 A372823 A367083 * A054188 A093368 A278025 Adjacent sequences: A271782 A271783 A271784 * A271786 A271787 A271788 KEYWORD nonn,easy AUTHOR R. J. Mathar, Apr 14 2016 STATUS approved

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Last modified August 9 01:26 EDT 2024. Contains 375024 sequences. (Running on oeis4.)