cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A077850 Expansion of (1-x)^(-1)/(1 - 2*x - x^2 + x^3).

Original entry on oeis.org

1, 3, 8, 19, 44, 100, 226, 509, 1145, 2574, 5785, 13000, 29212, 65640, 147493, 331415, 744684, 1673291, 3759852, 8448312, 18983186, 42654833, 95844541, 215360730, 483911169, 1087338528, 2443227496, 5489882352, 12335653673, 27717962203, 62281695728, 139945699987
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2002

Keywords

Comments

a(n) = A052534(n+1) - 1.

Links

Crossrefs

Cf. A188106. See also A189247. - L. Edson Jeffery, Apr 22 2011

Programs

  • Mathematica
    CoefficientList[Series[(1-x)^(-1)/(1-2x-x^2+x^3),{x,0,40}],x] (* or *) LinearRecurrence[{3,-1,-2,1},{1,3,8,19},40] (* Harvey P. Dale, Jan 22 2013 *)
  • PARI
    Vec(1/(1-x)/(1-2*x-x^2+x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

a(n) = sum(k=0..n+2, A006054(k)). - Philippe Deléham, Sep 07 2006
a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3) + a(n-4), n>3. Also a(n)=Sum_{k=0..n} A188106(n,k), n=0,1,2,..., giving partial sums of first convolution of A006054 with itself. - L. Edson Jeffery, Apr 22 2011

A189427 Expansion of (x^2)/((1-x)*(1-2*x-x^2+x^3)^2).

Original entry on oeis.org

0, 0, 1, 5, 19, 61, 180, 502, 1349, 3529, 9050, 22854, 57014, 140832, 345036, 839530, 2030757, 4887423, 11710757, 27951471, 66486128, 157661282, 372840407, 879510801, 2070045268, 4862121660, 11398688956, 26676792832, 62333380456, 145434747140
Offset: 0

Views

Author

L. Edson Jeffery, Apr 22 2011

Keywords

Comments

Second of a series of sequences of partial sums of (nonzero) diagonals of triangle A188106 whose diagonals correspond to successive convolutions of A006054 with itself, where the first such sequence of partial sums is given by A077850. For n=1,2,..., this series of sequences is generated by successive series expansion of 1/((1-x)*(1-2*x-x^2+x^3)^n), for which A077850 corresponds to n=1 and A189427 corresponds to n=2.
a(n)=Sum_{k=0..n} A189426(k), where A189426={0,0,1,4,14,42,119,322,...} is the convolution of A006054={0,0,1,2,5,11,25,56,126,...} with itself. Also, a(n+2)=Sum_{k=0..n} A188106{n+k+1,k}, n=0,1,2,....

Crossrefs

Cf. A006054, A077850, A188106, A189426.

Programs

Formula

G.f.: (x^2)/((1-x)*(1-2*x-x^2+x^3)^2).
a(n)=5*a(n-1)-6*a(n-2)-4*a(n-3)+9*a(n-4)-a(n-5)-3*a(n-6)+a(n-7), n>=7, a{m}={0,0,1,5,19,61,180}, m=0..6.
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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