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 A108488 Expansion of 1/sqrt(1 -2*x -3*x^2 -4*x^3 +4*x^4). 1
 1, 1, 3, 9, 23, 69, 203, 601, 1815, 5493, 16731, 51225, 157367, 485093, 1499499, 4646233, 14427095, 44880981, 139849979, 436419737, 1363713015, 4266417221, 13362194571, 41891406681, 131452430999, 412835452213, 1297543367835 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS In general, Sum_{k=0..n} C(n-k,k)^2*a^k*b^(n-k) has the expansion 1/sqrt(1 -2*b*x -(2*a*b -b^2)*x^2 -2*a*b^2*x^3 +(a*b)^2*x^4). LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 FORMULA a(n) = Sum_{k=0..n} C(n-k, k)^2*2^k. a(n) ~ ((4*sqrt(2)-1)/62)^(1/4) * (1+2*sqrt(2)+sqrt(1+4*sqrt(2)))^(n+1) /(sqrt(Pi*n)*2^(n+2)). - Vaclav Kotesovec, Jul 24 2013 D-finite with recurrence: n*a(n) +(-2*n+1)*a(n-1) +3*(-n+1)*a(n-2) +2*(-2*n+3)*a(n-3) +4*(n-2)*a(n-4)=0. - R. J. Mathar, Aug 06 2013 G.f.: exp( Sum_{n>=1} (x^n/n) * Sum_{k=0..n} C(2*n,2*k) * 2^k * x^k ). - Paul D. Hanna, Aug 31 2014 MATHEMATICA Table[Sum[Binomial[n-k, k]^2*2^k, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 24 2013 *) PROG (PARI) {a(n)=polcoeff( exp(sum(m=1, n, sum(k=0, m, binomial(2*m, 2*k) * 2^k * x^k) *x^m/m) +x*O(x^n)), n)} for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Aug 31 2014 CROSSREFS Cf. A051286, A108480. Sequence in context: A146661 A004666 A196488 * A018334 A146392 A116659 Adjacent sequences: A108485 A108486 A108487 * A108489 A108490 A108491 KEYWORD easy,nonn AUTHOR Paul Barry, Jun 04 2005 STATUS approved

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Last modified November 30 06:18 EST 2022. Contains 358431 sequences. (Running on oeis4.)