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 1-4 of 4 results.

A052973 Expansion of ( 1-x ) / ( 1-x-3*x^2+x^3 ).

Original entry on oeis.org

1, 0, 3, 2, 11, 14, 45, 76, 197, 380, 895, 1838, 4143, 8762, 19353, 41496, 90793, 195928, 426811, 923802, 2008307, 4352902, 9454021, 20504420, 44513581, 96572820, 209609143, 454814022, 987068631, 2141901554, 4648293425
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

References

  • Kenneth Edwards, Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, Part II, Fib. Q., 58:2 (2020), 169-177.

Links

Programs

  • Maple
    spec := [S,{S=Sequence(Prod(Union(Prod(Union(Z,Z),Sequence(Z)),Z),Z))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
  • Mathematica
    CoefficientList[Series[(1-x)/(1-x-3x^2+x^3),{x,0,30}],x] (* or *) LinearRecurrence[{1,3,-1},{1,0,3},40] (* Harvey P. Dale, Sep 06 2017 *)

Formula

G.f.: -(-1+x)/(1-x-3*x^2+x^3)
Recurrence: {a(1)=0, a(0)=1, a(2)=3, a(n)-3*a(n+1)-a(n+2)+a(n+3)=0}
Sum(-1/74*(1-34*_alpha+9*_alpha^2)*_alpha^(-1-n), _alpha=RootOf(1-_Z-3*_Z^2+_Z^3))
a(n) = A125691(n)-A125691(n-1). - R. J. Mathar, Feb 27 2019

Extensions

More terms from James Sellers, Jun 06 2000

A307464 Number of Catalan words of length n avoiding the pattern 000.

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 42, 90, 197, 425, 926, 2004, 4357, 9443, 20510, 44482, 96569, 209505, 454730, 986676, 2141361, 4646659, 10084066, 21882682, 47488221, 103052201, 223634182, 485302564, 1053152909, 2285426419, 4959582582, 10762708930, 23356030257, 50684574465
Offset: 0

Views

Author

R. J. Mathar, Apr 09 2019

Keywords

Links

Crossrefs

Cf. A000108, A125691, A356697, A356698.

Programs

  • Maple
    (1-2*x^2)/(1-x-3*x^2+x^3) ;
taylor(%,x=0,30) ;
gfun[seriestolist](%) ;
  • Mathematica
    LinearRecurrence[{1,3,-1},{1,1,2},40] (* Harvey P. Dale, Aug 06 2019 *)

Formula

a(n) = A125691(n)-2*A125691(n-2).
G.f.: (1-2*x^2)/(1-x-3*x^2+x^3).

A125690 Riordan array (1/(1-x-2x^2), x(1-x)/(1-x-2x^2)).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 11, 9, 7, 1, 1, 21, 25, 13, 9, 1, 1, 43, 53, 43, 17, 11, 1, 1, 85, 125, 97, 65, 21, 13, 1, 1, 171, 273, 255, 153, 91, 25, 15, 1, 1, 341, 609, 597, 441, 221, 121, 29, 17, 1, 1, 683, 1325, 1443
Offset: 0

Views

Author

Paul Barry, Nov 30 2006

Keywords

Comments

First column is A001045. Row sums are Pell numbers A000129(n+1). Diagonal sums are A125691.

Examples

			Triangle begins
1,
1, 1,
3, 1, 1,
5, 5, 1, 1,
11, 9, 7, 1, 1,
21, 25, 13, 9, 1, 1,
43, 53, 43, 17, 11, 1, 1

Formula

G.f.: 1/(1-x-2x^2-xy(1-x)).
T(n,k) = T(n-1,k) + T(n-1,k-1) + 2*T(n-2,k) - T(n-2,k-1). - Philippe Deléham, Feb 24 2012

A208153 Convolution triangle based on A006053.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 4, 7, 3, 1, 9, 14, 12, 4, 1, 14, 35, 31, 18, 5, 1, 28, 70, 87, 56, 25, 6, 1, 47, 154, 207, 175, 90, 33, 7, 1, 89, 306, 504, 476, 310, 134, 42, 8, 1, 155, 633, 1137, 1274, 941, 504, 189, 52, 9, 1
Offset: 0

Views

Author

Philippe Deléham, Feb 24 2012

Keywords

Comments

Riordan array (1/(1-x-2*x^2+x^3), x/(1-x-2*x^2+x^3)).
Subtriangle of triangle given by (0, 1, 2, -5/2, 1/10, 2/5, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Diagonal sums are A125691(n).
Row sums are A001654(n+1).
Mirror image of triangle in A188107.

Examples

			Triangle begins:
  1
  1, 1
  3, 2, 1
  4, 7, 3, 1
  9, 14, 12, 4, 1
  14, 35, 31, 18, 5, 1
Triangle (0, 1, 2, -5/2, 1/10, 2/5, 0, 0,...) DELTA (1, 0, 0, 0,...) begins:
  1
  0, 1
  0, 1, 1
  0, 3, 2, 1
  0, 4, 7, 3, 1
  0, 9, 14, 12, 4, 1
  0, 14, 35, 31, 18, 5, 1

Links

Crossrefs

Cf. A006053, A000012, A000027, A055998.

Programs

  • Mathematica
    nmax=9; Flatten[CoefficientList[Series[CoefficientList[Series[1/(1 - x - 2*x^2 + x^3 - y*x), {x, 0, nmax}], x], {y, 0, nmax}], y]] (* Indranil Ghosh, Mar 10 2017 *)

Formula

T(n,k) = T(n-1,k-1) + T(n-1,k) + 2*T(n-2,k) - T(n-3,k).
G.f.: 1/(1-x-2*x^2+x^3-y*x).
Sum_{k>=0} T(n-2*k,k) = A001045(n+1).
Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A008346(n), A006053(n+2), A001654(n+1) for x = -1, 0, 1 respectively.
Showing 1-4 of 4 results.

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

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