A270386 Expansion of (4/(3*x/(1-x))) * sin((1/3)*arcsin(sqrt(27*x/4/(1-x))))^2.

1, 2, 9, 46, 256, 1510, 9283, 58848, 381963, 2525916, 16958498, 115288674, 792042589, 5490312864, 38352695246, 269719400974, 1908059370583, 13568804436340, 96942782340802, 695513575242284, 5008808999633736, 36195063931874308, 262372258663337954

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{k=0..n}(binomial(n-1,n-k)*((binomial(3*k+1,k+1))/(2*k+1))).

G.f.: g(x/(1-x)) where g(x) is the g.f. of A006013.

a(n) ~ 31^(n + 1/2) / (sqrt(Pi) * n^(3/2) * 2^(2*n + 2)). - Vaclav Kotesovec, Mar 16 2016

a(n) = 2*hypergeometric([5/3, 7/3, 1-n],[5/2, 3], -27/4)) for n>0. - Peter Luschny, Mar 16 2016

Conjecture D-finite with recurrence: 2*(n+1)*(2*n+1)*a(n) +(-39*n^2+8*n+5)*a(n-1) +(66*n-37)*(n-2)*a(n-2) -31*(n-2)*(n-3)*a(n-3)=0. - R. J. Mathar, Jun 07 2016

Mathematica

Table[Sum[(Binomial[n - 1, n - k]*((Binomial[3*k + 1, k + 1])/(2*k + 1))), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 16 2016, after Vladimir Kruchinin *)

Prog

(Maxima)
a(n):=(sum(binomial(n-1, n-k)*((binomial(3*k+1, k+1))/(2*k+1)), k, 0, n));
(PARI) a(n) = sum(k=0, n, binomial(n-1, n-k)*((binomial(3*k+1, k+1))/ (2*k+1))); \\ Michel Marcus, Mar 16 2016
(Sage)
a = lambda n: simplify(2*hypergeometric([5/3, 7/3, 1-n], [5/2, 3], -27/4)) if n>0 else 1
[a(n) for n in range(23)] # Peter Luschny, Mar 16 2016

Crossrefs

Cf. A006013.

Sequence in context: A036726 A219197 A340942 * A181997 A020053 A114194

Adjacent sequences: A270383 A270384 A270385 * A270387 A270388 A270389

Keyword

nonn

Author

Vladimir Kruchinin, Mar 16 2016

Status

approved