A190667 - OEIS

%I #24 Feb 25 2023 12:08:33

%S 1,3,5,13,30,69,160,371,859,1990,4610,10679,24738,57306,132750,307517,

%T 712367,1650207,3822725,8855390,20513621,47520058,110080805,255003553,

%U 590718900,1368407674,3169933385,7343190086,17010591104,39405245720

%N Expansion of (1+2*x)/(1-x^4-2*x^3-2*x^2-x).

%H Harvey P. Dale, <a href="/A190667/b190667.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,2,1).

%F a(n) = (n+1)*sum(k=1..n+1, binomial(k, n-k+1)*A000045(k)/k).

%F a(n) = a(n-1)+2*a(n-2)+2*a(n-3)+a(n-4), a(0)=1, a(1)=3, a(2)=5, a(3)=13.

%t CoefficientList[Series[(1+2x)/(1-x^4-2x^3-2x^2-x),{x,0,40}],x] (* or *) LinearRecurrence[ {1,2,2,1},{1,3,5,13},40] (* _Harvey P. Dale_, Feb 25 2023 *)

%o (Maxima) a(n):=(n+1)*sum(binomial(k,n-k+1)*fib(k)/k,k,1,n+1);makelist(a(n),n,0,35);

%o (Maxima) a(n):=if n<0 then 0 else if n=0 then 1 else if n=1 then 3 else if n=2 then 5 else if n=3 then 13 else a(n-1)+2*a(n-2)+2*a(n-3)+a(n-4);

%o (Magma) [ (n+1)*&+[ (Binomial(k, n-k+1)*Fibonacci(k))/k: k in [1..n+1] ]: n in [0..35] ]; // _Klaus Brockhaus_, May 17 2011

%o (Magma) [ n eq 1 select 1 else n eq 2 select 3 else n eq 3 select 5 else n eq 4 select 13 else Self(n-1)+2*Self(n-2)+2*Self(n-3)+Self(n-4): n in [1..36] ]; // _Klaus Brockhaus_, Jun 01 2011

%K nonn

%O 0,2

%A _Dmitry Kruchinin_, May 16 2011