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A054109
a(n) = T(2*n+1, n), array T as in A054106.
4
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
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
KEYWORD
nonn
STATUS
approved