A106191 Expansion of sqrt(1-4x)/(1-x).

1, -1, -3, -7, -17, -45, -129, -393, -1251, -4111, -13835, -47427, -164999, -581023, -2066823, -7415703, -26805393, -97520733, -356810313, -1312087713, -4846614093, -17974854933, -66907388973, -249872516253, -935991743553, -3515800038201, -13239692841105

Offset

0,3

Comments

Row sums of number triangle A106190. Partial sums of A002420.

For n >= 1, the absolute values also give the iterates of A122237, starting from 0. (A122237(0), A122237(A122237(0)), A122237(A122237(A122237(0))), ...), this stems from the fact that the sequence gives the positions of terms with binary expansion 1(10){n-1}0 in A014486 (see A080675).

Links

Table of n, a(n) for n=0..26.

Formula

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

G.f.: (2/(1-x))/G(0), where G(k) = 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013

D-finite with recurrence: a(0)=1, a(1)=-1; for n>1, a(n) = (1/n)*((5*n-6)*a(n-1) - (4*n-6)*a(n-2)). - Tani Akinari, Aug 25 2013

Crossrefs

|a(n)| = A080300(A080675(n)) = A075161(A001348(n)) (for n >= 1) = A075163(A000244(A008578(n-2))) = A014137(n-1)+A014138(n-2) = 2*A014137(n-1)-1, for n >= 2 (because binomial(2n+2, n+1)/(2n+1) = 2*A000108(n)).

Sequence in context: A261235 A211281 A233656 * A062810 A113985 A309302

Adjacent sequences: A106188 A106189 A106190 * A106192 A106193 A106194

Keyword

easy,sign

Author

Paul Barry, Apr 24 2005

Extensions

Barry's formula made more succinct, as well as comments regarding interpretation as absolute values added by Antti Karttunen, Sep 14 2006

Status

approved