A054109 a(n) = T(2*n+1, n), array T as in A054106.

1, 2, 8, 27, 99, 363, 1353, 5082, 19228, 73150, 279566, 1072512, 4127788, 15930512, 61628248, 238911947, 927891163, 3609676487, 14062955413, 54860308997, 214268628223, 837780853637, 3278934510163, 12844867331387

Offset

0,2

Comments

Hankel transform of A054109. - Paul Barry, Nov 04 2009

From Paul Barry, Mar 29 2010: (Start)

Hankel transform is A167478 (correction of previous entry).

The aerated sequence 0,0,1,0,2,0,8,0,... has e.g.f. Integral_{t=0..x} cos(x-t)*Bessel_I(1,2t). (End)

Hankel transform of 0,1,2,8,27,... is -F(2n). - Paul Barry, Jan 17 2020

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

a(n-1) = (1/2)*(-1)^n*Sum_{k=1..n} (-1)^k*binomial(2k, k). - Benoit Cloitre, Nov 07 2002

Conjecture: (n+1)*a(n) + (-3*n-1)*a(n-1) + 2*(-2*n-1)*a(n-2) = 0. - R. J. Mathar, Nov 24 2012

a(n) ~ 2^(2*n+3) / (5*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 12 2014

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(2k+1,k+1). - Paul Barry, Jan 17 2020

G.f.: c(x)B(x)/(1+x), c(x) g.f. of A000108, B(x) g.f. of A000984. - Paul Barry, Jan 17 2020

a(n) = binomial(2*n+3, n+2)*hypergeom([1, n+5/2], [n+3], -4) + (-1)^n*(5 - sqrt(5)) /10. - Peter Luschny, Jan 18 2020

Maple

a := n -> abs(add(binomial(-j-1, -2*j-2), j=0..n)):
seq(a(n), n=0..23); # Zerinvary Lajos, Oct 03 2007
gf := ((1 - 4*x)^(-1/2) - 1)/(2*x*(x + 1)): ser := series(gf, x, 32):
seq(coeff(ser, x, n), n=0..23); # Peter Luschny, Jan 18 2020

Mathematica

Table[FullSimplify[1/2*(-1)^(1+n) * (-1+1/Sqrt[5]-(-1)^n*Binomial[2*(2+n), 2+n] * Hypergeometric2F1[1, 5/2+n, 3+n, -4])], {n, 0, 20}] (* Vaclav Kotesovec, Feb 12 2014 *)
Table[1/2*(-1)^(n+1)*Sum[(-1)^k*Binomial[2*k, k], {k, 1, n+1}], {n, 0, 20}] (* Vaclav Kotesovec, Feb 12 2014 *)

Prog

(PARI) a(n)=(1/2)*(-1)^(n+1)*sum(k=1, n+1, (-1)^k*binomial(2*k, k))

Crossrefs

Cf. A054106, A000108, A000984.

Sequence in context: A138388 A138386 A096647 * A305256 A323613 A150710

Adjacent sequences: A054106 A054107 A054108 * A054110 A054111 A054112

Keyword

nonn

Author

Clark Kimberling

Status

approved