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 A262664 Expansion of (1-2*x)/((2-x)*sqrt(5*x^2-6*x+1))+1/(2-x). 1
 1, 1, 3, 13, 59, 271, 1257, 5881, 27715, 131395, 626033, 2995147, 14380181, 69249337, 334345091, 1617924973, 7844900339, 38105139907, 185380469961, 903147125143, 4405621159969, 21515837558557, 105188202097091, 514747668977263 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 FORMULA a(n) = Sum_{k=0..n}(binomial(n,k)*Sum_{i=0..n-k}(2^i*binomial(k,n-k-i)*binomial(k+i-1,i)*(-1)^(n-k-i))). G.f.: A(x) = x*B'(x)/B(x), where B(x)/x is g.f. of A033321. a(n) ~ 5^(n+1/2)/(6*sqrt(Pi*n)). - Vaclav Kotesovec, Sep 29 2015 D-finite with recurrence: 2*n*(3*n-4)*a(n) = (39*n^2 - 70*n + 28)*a(n-1) - (48*n^2 - 103*n + 34)*a(n-2) + 5*(n-2)*(3*n-1)*a(n-3). - Vaclav Kotesovec, Sep 29 2015 MATHEMATICA Table[Sum[Binomial[n, k] Sum[2^i Binomial[k, n - k - i] Binomial[k + i - 1, i] (-1)^(n - k - i), {i, 0, n - k}], {k, 0, n}], {n, 0, 23}] (* Michael De Vlieger, Sep 26 2015 *) PROG (Maxima) a(n):=sum(binomial(n, k)*sum(2^i*binomial(k, n-k-i)*binomial(k+i-1, i)*(-1)^(n-k-i), i, 0, n-k), k, 0, n); (PARI) x='x+O('x^50); Vec((1-2*x)/((2-x)*sqrt(5*x^2-6*x+1))+1/(2-x)) \\ G. C. Greubel, Jun 04 2017 CROSSREFS Cf. A033321. Sequence in context: A199297 A152594 A052984 * A151229 A151230 A151231 Adjacent sequences:  A262661 A262662 A262663 * A262665 A262666 A262667 KEYWORD nonn AUTHOR Vladimir Kruchinin, Sep 26 2015 STATUS approved

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Last modified April 4 07:32 EDT 2020. Contains 333213 sequences. (Running on oeis4.)