A055989 -id:A055989 - cp's OEIS frontend

A055991 a(n) is its own 4th difference.

1, 5, 19, 69, 250, 907, 3292, 11949, 43371, 157422, 571388, 2073943, 7527704, 27322992, 99173120, 359964521, 1306548149, 4742323107, 17213011605, 62477347458, 226771411939, 823102698260, 2987581397893, 10843899100203

Offset: 1

Henry Bottomley, Jun 02 2000

a(n) is the number of distinct matrix products in (A+B+C+D+E)^n where A,B,C and D all commute with each other, but not with E. - Paul D. Hanna and Max Alekseyev, Feb 01 2006
Row sums of Riordan array (1,1/(1-x)^4). - Paul Barry, Feb 02 2006
Quadrisection of A003269: a(n)=A003269(4n-1). - Paul Barry, Feb 02 2006
From Gary W. Adamson, Apr 23 2009: (Start)
Equals the INVERT transform of the tetrahedral series.
a(4) = 69 = (1, 4, 10) dot (19, 5, 1) + 20; = (19 + 20 + 10) + 20. (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, example 16.
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
Index entries for linear recurrences with constant coefficients, signature (5,-6,4,-1).

Cf. A055988, A055989, A055990 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.

I:=[1, 5, 19, 69]; [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

LinearRecurrence[{5,-6,4,-1},{1,5,19,69},30] (* Harvey P. Dale, Feb 27 2013 *)

a(n) = 5*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) = a(n-1)+A055990(n) = A055988(n+1)-A055988(n) = A055989(n+1)-2*A055989(n)+A055989(n-1).
Letting a(0)=1, we have a(n)=sum(u=0, n-1, sum(v=0, u, sum(w=0, v, sum(x=0, w, a(x))))) for n>0. - Benoit Cloitre, Jan 26 2003
a(n) = sum_{k=1..n} binomial(n+3*k-1, n-k). - Vladeta Jovovic, Mar 23 2003
a(n) = sum{k=0..n, binomial(4n-3k-1,k)}. - Paul Barry, Feb 02 2006
G.f.: x/(1-5x+6x^2-4x^3+x^4). - Paul Barry, Feb 02 2006

A055988 Sequence is its own 4th difference.

Original entry on oeis.org

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

Henry Bottomley, Jun 02 2000

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

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

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.

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

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 *)

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)

More terms from James Sellers, Jun 05 2000

A055990 a(n) is its own 4th difference.

Original entry on oeis.org

1, 4, 14, 50, 181, 657, 2385, 8657, 31422, 114051, 413966, 1502555, 5453761, 19795288, 71850128, 260791401, 946583628, 3435774958, 12470688498, 45264335853, 164294064481, 596331286321, 2164478699633, 7856317702310, 28515747394555, 103502414271126

Offset: 1

Henry Bottomley, Jun 02 2000

Number of compositions of 4*n-2 into parts 1 and 4. - Seiichi Manyama, Feb 03 2024

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

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.

Cf. A003269.

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

CoefficientList[Series[(1-x)/(1-5*x+6*x^2-4*x^3+x^4),{x,0,30}],x] (* Vincenzo Librandi, Apr 06 2012 *)
LinearRecurrence[{5,-6,4,-1},{1,4,14,50},30] (* Harvey P. Dale, Oct 18 2015 *)

a(n):=sum((binomial(n+3*m+1,n-m-1)),m,0,n-1); /* Vladimir Kruchinin, Nov 18 2020 */

Vec((1-x)/(1-5*x+6*x^2-4*x^3+x^4)+O(x^99)) \\ Charles R Greathouse IV, Apr 06 2012

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).
G.f.: x*(1-x)/(1-5*x+6*x^2-4*x^3+x^4). [Colin Barker, Apr 05 2012]
a(n) = Sum_{m=0..n-1} C(n+3m+1,n-m-1). - Vladimir Kruchinin, Nov 18 2020

More terms from James Sellers, Jun 05 2000

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A055991 a(n) is its own 4th difference.

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A055988 Sequence is its own 4th difference.

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A055990 a(n) is its own 4th difference.

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