login
Expansion of g.f. (1+x)^6/(1-x).
7

%I #27 Mar 19 2023 09:44:49

%S 1,7,22,42,57,63,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,

%T 64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,

%U 64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64

%N Expansion of g.f. (1+x)^6/(1-x).

%C a(n)=2^6=64 for n>=6. We observe that this sequence is the transform of A171440 by T such that: T(u_0,u_1,u_2,u_3,u_4,u_5,...)=(u_0,u_0+u_1,u_1+u_2,u_2+u_3,u_3+u_4,...).

%C Also continued fraction expansion of 1+(1233212607598+5*sqrt(41))/8688482797079. - _Bruno Berselli_, Sep 23 2011

%H Richard Choulet, <a href="https://mp.sbpm.be/MP157.PDF">Une nouvelle formule de combinatoire?</a>, Mathématique et Pédagogie, 157 (2006), p. 53-60. In French.

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).

%F With m=7, a(n) = Sum_{k=0..floor(n/2)} binomial(m,n-2*k).

%e a(4) = C(7,4-0) + C(7,4-2) + C(7,4-4) = 35+21+1 = 57.

%p m:=7:for n from 0 to 40 do a(n):=sum('binomial(m,n-2*k)',k=0..floor(n/2)): od : seq(a(n),n=0..40);

%Y Cf. A040000, A113311, A115291, A171418, A171440, A171442, A171443.

%K nonn,easy

%O 0,2

%A _Richard Choulet_, Dec 09 2009

%E Definition rewritten by _Bruno Berselli_, Sep 23 2011