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a(n) is its own 4th difference.
6

%I #36 Sep 07 2024 15:39:15

%S 1,4,14,50,181,657,2385,8657,31422,114051,413966,1502555,5453761,

%T 19795288,71850128,260791401,946583628,3435774958,12470688498,

%U 45264335853,164294064481,596331286321,2164478699633,7856317702310,28515747394555,103502414271126

%N a(n) is its own 4th difference.

%C Number of compositions of 4*n-2 into parts 1 and 4. - _Seiichi Manyama_, Feb 03 2024

%H Vincenzo Librandi, <a href="/A055990/b055990.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6,4,-1).

%F a(n) = 5*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) = a(n-1)+A055989(n) = A055991(n)-A055991(n-1) = A055988(n+1)-2*A055988(n)+A055988(n-1).

%F G.f.: x*(1-x)/(1-5*x+6*x^2-4*x^3+x^4). [_Colin Barker_, Apr 05 2012]

%F a(n) = Sum_{m=0..n-1} C(n+3m+1,n-m-1). - _Vladimir Kruchinin_, Nov 18 2020

%t CoefficientList[Series[(1-x)/(1-5*x+6*x^2-4*x^3+x^4),{x,0,30}],x] (* _Vincenzo Librandi_, Apr 06 2012 *)

%t LinearRecurrence[{5,-6,4,-1},{1,4,14,50},30] (* _Harvey P. Dale_, Oct 18 2015 *)

%o (Magma) I:=[1, 4, 14, 50]; [n le 4 select I[n] else 5*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Apr 06 2012

%o (PARI) Vec((1-x)/(1-5*x+6*x^2-4*x^3+x^4)+O(x^99)) \\ _Charles R Greathouse IV_, Apr 06 2012

%o (Maxima)

%o a(n):=sum((binomial(n+3*m+1,n-m-1)),m,0,n-1); /* _Vladimir Kruchinin_, Nov 18 2020 */

%Y Cf. A055988, A055989, A055991 for the other differences of a(n). See A000079, A001906, A052529 for examples of sequences which are respectively their own first, second and third differences.

%Y Cf. A003269.

%K nonn,easy

%O 1,2

%A _Henry Bottomley_, Jun 02 2000

%E More terms from _James A. Sellers_, Jun 05 2000