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A059278 G.f. is G(x*(1-x)/(1-2*x)) where G(x) is g.f. for Catalan numbers A000108. 3
1, 1, 3, 11, 43, 175, 735, 3167, 13935, 62383, 283311, 1302271, 6047679, 28332991, 133752191, 635618431, 3038326911, 14599154431, 70474889471, 341624867071, 1662254107391, 8115717976831, 39747223425791, 195219110182911, 961330824858623 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Robert Israel, Table of n, a(n) for n = 0..1000

FORMULA

G.f.: (2*x-1+sqrt((1-2*x)*(1-6*x+4*x^2)))/(2*x*(x-1)).

G.f.: W(0), where W(k) = 1 + (4*k+1)*x*(1-x)/( (k+1)*(1-2*x) - 2*x*(1-x)*(1-2*x)*(k+1)*(4*k+3)/(2*x*(1-x)*(4*k+3) + (2*k+3)*(1-2*x)/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 28 2013

Recurrence: (n+1)*a(n) = 3*(3*n-1)*a(n-1) - 12*(2*n-3)*a(n-2) + 12*(2*n-5)*a(n-3) - 4*(2*n-7)*a(n-4). - Vaclav Kotesovec, Jun 19 2014

a(n) ~ sqrt(10-2*sqrt(5)) * (3+sqrt(5))^n / (2*sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 19 2014

a(n) = Sum_{k=0..n}((C(k)*Sum_{i=0..n-k}(2^i*binomial(k,n-k-i)*binomial(k+i-1,i)*(-1)^(n-k-i))))), where C(k) is Catalan numbers. - Vladimir Kruchinin, Mar 04 2016

MAPLE

f:= gfun:-rectoproc({(4+8*n)*a(n)+(-36-24*n)*a(1+n)+(60+24*n)*a(n+2)+(-33-9*n)*a(n+3)+(5+n)*a(n+4), a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 11}, a(n), remember):

map(f, [$0..30]); # Robert Israel, Mar 04 2016

MATHEMATICA

CoefficientList[Series[(2*x-1+Sqrt[(1-2*x)*(1-6*x+4*x^2)])/(2*x*(x-1)), {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 19 2014 *)

Table[Sum[CatalanNumber[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, 24}] (* Michael De Vlieger, Mar 04 2016 *)

PROG

(PARI) a(n)=polcoeff((2*x-1 +sqrt((1-2*x)*(1-6*x+4*x^2)+x^2*O(x^n))) /(2*x^2-2*x), n);

(PARI) x='x+O('x^100); Vec((2*x-1+sqrt((1-2*x)*(1-6*x+4*x^2)))/(2*x*(x-1))) \\ Altug Alkan, Mar 05 2016

(Maxima)

a(n):=sum((binomial(2*k, k)*sum(2^i*binomial(k, n-k-i)*binomial(k+i-1, i)*(-1)^(n-k-i), i, 0, n-k))/(k+1), k, 0, n); /* Vladimir Kruchinin, Mar 04 2016 */

CROSSREFS

Cf. A000108.

Sequence in context: A026876 A270447 A151090 * A151091 A151092 A151093

Adjacent sequences:  A059275 A059276 A059277 * A059279 A059280 A059281

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jan 24 2001

STATUS

approved

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Last modified June 23 11:24 EDT 2021. Contains 345397 sequences. (Running on oeis4.)