A102592 a(n) = Sum_{k=0..n} binomial(2n+1, 2k)*5^(n-k).

1, 8, 80, 832, 8704, 91136, 954368, 9994240, 104660992, 1096024064, 11477712896, 120196169728, 1258710630400, 13181388849152, 138037296103424, 1445545331654656, 15137947242201088, 158526641599938560

Offset

0,2

Comments

In general, Sum_{k=0..n} binomial(2n+1,2k)*r^(n-k) has g.f. (1-(r-1)x)/(1-2(r+1)+(r-1)^2x^2) and a(n) = ((sqrt(r)-1)^(2n+1) + (sqrt(r)+1)^(2n+1))/(2*sqrt(r)).

Links

Table of n, a(n) for n=0..17.

Index entries for linear recurrences with constant coefficients, signature (12,-16).

Formula

G.f.:(1-4x)/(1-12x+16x^2);

a(n) = 12*a(n-1) - 16*a(n-2);

a(n) = sqrt(5)*(sqrt(5)-1)^(2n+1)/10 + sqrt(5)*(sqrt(5)+1)^(2n+1)/10.

a(n) = Sum_{k=0..n} binomial(2n+1, k+1)*5^k. - Paul Barry, May 27 2005

a(n) = 4^(n+1)*A001519(n+1). - N. J. A. Sloane, Apr 13 2011

a(n) = 5^n* 2F1(-n-1/2, -n ; 1/2 ; 1/5). - R. J. Mathar, Aug 23 2024

Crossrefs

Cf. A066443, A102591.

Sequence in context: A299871 A136949 A346178 * A345081 A320759 A233123

Adjacent sequences: A102589 A102590 A102591 * A102593 A102594 A102595

Keyword

easy,nonn

Author

Paul Barry, Jan 22 2005

Status

approved