A055988 Sequence is its own 4th difference.

1, 2, 7, 26, 95, 345, 1252, 4544, 16493, 59864, 217286, 788674, 2862617, 10390321, 37713313, 136886433, 496850954, 1803399103, 6545722210, 23758733815, 86236081273, 313007493212, 1136110191472, 4123691589365, 14967590689568

Offset: 1

Author

Henry Bottomley, Jun 02 2000

Keywords

• nonn
• easy

Comments

Row sums of Riordan array (1/(1-x), x/(1-x)^4), A109960. - Paul Barry, Jul 06 2005

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-6,4,-1).

Crossrefs

Cf. A055989, A055990, 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.

Programs

Magma

I:=[1, 2, 7, 26]; [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 05 2012

Mathematica

CoefficientList[Series[(1-x)^3/(1-5x+6x^2-4x^3+x^4),{x,0,40}],x] (* Vincenzo Librandi, Apr 05 2012 *)
LinearRecurrence[{5,-6,4,-1},{1,2,7,26},30] (* Harvey P. Dale, Jan 15 2017 *)

Formula

a(n) = 5a(n-1) - 6a(n-2) + 4a(n-3) - a(n-4) = a(n-1) + A055991(n-1) = A055989(n) - A055989(n-1) = A055990(n) - 2*A055990(n-1) + A055990(n-2).

From Paul Barry, Jul 06 2005: (Start)

G.f.: (1-x)^3/(1 - 5x + 6x^2 - 4x^3 + x^4);

a(n) = Sum_{k=0..n} binomial(n+3k, 4k). (End)

Extensions

More terms from James Sellers, Jun 05 2000