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 1-3 of 3 results.

A026671 Number of lattice paths from (0,0) to (n,n) with steps (0,1), (1,0) and, when on the diagonal, (1,1).

Original entry on oeis.org

1, 3, 11, 43, 173, 707, 2917, 12111, 50503, 211263, 885831, 3720995, 15652239, 65913927, 277822147, 1171853635, 4945846997, 20884526283, 88224662549, 372827899079, 1576001732485, 6663706588179, 28181895551161, 119208323665543, 504329070986033, 2133944799315027
Offset: 0

Views

Author

Clark Kimberling, Miklos Bona

Keywords

Comments

1, 1, 3, 11, 43, 173, ... is the unique sequence for which both the Hankel transform of the sequence itself and the Hankel transform of its left shift are the powers of 2 (A000079). For example, det[{{1, 1, 3}, {1, 3, 11}, {3, 11, 43}}] = det[{{1, 3, 11}, {3, 11, 43}, {11, 43, 173}}] = 4. - David Callan, Mar 30 2007
From Paul Barry, Jan 25 2009: (Start)
a(n) is the image of F(2n+2) under the Catalan matrix (1,xc(x)) where c(x) is the g.f. of A000108.
The sequence 1,1,3,... is the image of A001519 under (1,xc(x)). This sequence has g.f. given by 1/(1-x-2x^2/(1-3x-x^2/(1-2x-x^2/(1-2x-x^2/(1-... (continued fraction). (End)
Binomial transform of A111961. - Philippe Deléham, Feb 11 2009
From Paul Barry, Nov 03 2010: (Start)
The sequence 1,1,3,... has g.f. 1/(1-x/sqrt(1-4x)), INVERT transform of A000984.
It is an eigensequence of the sequence array for A000984. (End)

References

  • L. W. Shapiro and C. J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 33-46.

Links

Crossrefs

a(n) = T(2n-1, n-1), T given by A026736.
a(n) = T(2n, n), T given by A026670.
a(n) = T(2n+1, n+1), T given by A026725.
Row sums of triangle A054335.
Cf. A026781, A001076, A176280.

Programs

  • GAP
    a:=[3,11,43];; for n in [4..30] do a[n]:=(2*(4*n-3)*a[n-1] - 3*(5*n-8)*a[n-2] - 2*(2*n-3)*a[n-3])/n; od; Concatenation([1], a); # G. C. Greubel, Jul 16 2019
  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/(Sqrt(1-4*x)-x) )); // G. C. Greubel, Jul 16 2019
  • Mathematica
    Table[SeriesCoefficient[1/(Sqrt[1-4*x]-x),{x,0,n}],{n,0,30}] (* Vaclav Kotesovec, Oct 08 2012 *)
  • PARI
    {a(n)= if(n<0, 0, polcoeff( 1/(sqrt(1 -4*x +x*O(x^n)) -x), n))} /* Michael Somos, Apr 20 2007 */
  • PARI
    my(x='x+O('x^66)); Vec( 1/(sqrt(1-4*x)-x) ) \\ Joerg Arndt, May 04 2013
  • Sage
    (1/(sqrt(1-4*x)-x)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 16 2019

Formula

From Wolfdieter Lang, Mar 21 2000: (Start)
G.f.: 1/(sqrt(1-4*x)-x).
a(n) = Sum_{i=1..n} a(i-1)*binomial(2*(n-i), n-i) + binomial(2*n, n), n >= 1, a(0)=1. (End)
G.f.: 1/(1 -x -2*x*c(x)) where c(x) = g.f. for Catalan numbers A000108. - Michael Somos, Apr 20 2007
From Paul Barry, Jan 25 2009: (Start)
G.f.: 1/(1 - 3xc(x) + x^2*c(x)^2);
G.f.: 1/(1-3x-2x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-... (continued fraction).
a(0) = 1, a(n) = Sum_{k=0..n} (k/(2n-k))*C(2n-k,n-k)*F(2k+2). (End)
a(n) = Sum_{k=0..n} A039599(n,k) * A000045(k+2). - Philippe Deléham, Feb 11 2009
From Paul Barry, Feb 08 2009: (Start)
G.f.: 1/(1-x/(1-2x/(1-x/(1-x/(1-x/(1-x/(1-x/(1-... (continued fraction);
G.f. of 1,1,3,... is 1/(1-x-2x/(1-x/(1-x/(1-x/(1-... (continued fraction). (End)
From Gary W. Adamson, Jul 14 2011: (Start)
a(n) = the upper left term in M^n, M = the infinite square production matrix:
3, 2, 0, 0, 0, 0, ...
1, 1, 1, 0, 0, 0, ...
1, 1, 1, 1, 0, 0, ...
1, 1, 1, 1, 1, 0, ...
1, 1, 1, 1, 1, 1, ...
... (End)
From Vaclav Kotesovec, Oct 08 2012: (Start)
D-finite with recurrence: n*a(n) = 2*(4*n-3)*a(n-1) - 3*(5*n-8)*a(n-2) - 2*(2*n-3)*a(n-3).
a(n) ~ (2+sqrt(5))^n/sqrt(5). (End)
a(n) = Sum_{k=0..n+1} 4^(n+1-k) * binomial(n-k/2,n+1-k). - Seiichi Manyama, Mar 30 2025
From Peter Luschny, Mar 30 2025: (Start)
a(n) = 4^n*(binomial(n-1/2, n)*hypergeom([1, (1-n)/2, -n/2], [1/2, 1/2-n], -1/4) + hypergeom([(1-n)/2, 1-n/2], [1-n], -1/4)/4) for n > 0.
a(n) = A001076(n) + A176280(n). (End)

A111960 Renewal array for central trinomial numbers A002426.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 7, 7, 3, 1, 19, 20, 12, 4, 1, 51, 61, 40, 18, 5, 1, 141, 182, 135, 68, 25, 6, 1, 393, 547, 441, 251, 105, 33, 7, 1, 1107, 1640, 1428, 888, 420, 152, 42, 8, 1, 3139, 4921, 4572, 3076, 1596, 654, 210, 52, 9, 1, 8953, 14762, 14535, 10456, 5880, 2652, 966, 280, 63, 10, 1
Offset: 0

Views

Author

Paul Barry, Aug 23 2005

Keywords

Comments

Also the convolution triangle of A002426. - Peter Luschny, Oct 06 2022

Examples

			Triangle T(n,k) begins:
   1;
   1,  1;
   3,  2,  1;
   7,  7,  3,  1;
  19, 20, 12,  4, 1;
  51, 61, 40, 18, 5, 1;
  ...
From _Paul Barry_, May 12 2009: (Start)
Production matrix is
  1, 1,
  2, 1, 1,
  0, 2, 1, 1,
  -2, 0, 2, 1, 1,
  0, -2, 0, 2, 1, 1,
  4, 0, -2, 0, 2, 1, 1. (End)

Crossrefs

Row sums are A111961.
Diagonal sums are A111962.
Inverse is A111963.
Factors as A007318*A111959.
Column k=0 gives A002426.
Cf. A026325.

Programs

  • Maple
    # Uses function PMatrix from A357368. Adds a row and column above and to the left.
PMatrix(10, n -> A002426(n - 1)); # Peter Luschny, Oct 06 2022

Formula

Factors as (1/(1-x), x/(1-x))*(1/sqrt(1-4x^2), x/sqrt(1-4x^2)).
From Paul Barry, May 12 2009: (Start)
Equals ((1-x^2)/(1+x+x^2),x/(1+x+x^2))^{-1}*(1,x/(1-x^2))=A094531*(1,x/(1-x^2)).
Riordan array (1/sqrt(1-2x-3x^2), x/sqrt(1-2x-3x^2));
T(n,k) = Sum_{j=0..n} C(n,j)*C((j-1)/2,(j-k)/2)*2^(j-k)*(1+(-1)^(j-k))/2.
G.f.: 1/(1-xy-x-2x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-... (continued fraction). (End)

A321621 The Riordan square of the Motzkin numbers, triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 4, 8, 5, 1, 9, 21, 18, 7, 1, 21, 55, 58, 32, 9, 1, 51, 145, 177, 123, 50, 11, 1, 127, 385, 525, 431, 224, 72, 13, 1, 323, 1030, 1532, 1429, 889, 369, 98, 15, 1, 835, 2775, 4428, 4572, 3269, 1639, 566, 128, 17, 1
Offset: 0

Views

Author

Peter Luschny, Nov 22 2018

Keywords

Examples

			[0][   1]
[1][   1,    1]
[2][   2,    3,    1]
[3][   4,    8,    5,    1]
[4][   9,   21,   18,    7,    1]
[5][  21,   55,   58,   32,    9,    1]
[6][  51,  145,  177,  123,   50,   11,   1]
[7][ 127,  385,  525,  431,  224,   72,  13,   1]
[8][ 323, 1030, 1532, 1429,  889,  369,  98,  15,  1]
[9][ 835, 2775, 4428, 4572, 3269, 1639, 566, 128, 17, 1]

Links

Crossrefs

T(n, 0) = A001006 (Motzkin), A111961 (row sums), A000007 (alternating row sums).
Cf. A321620.

Programs

  • Maple
    # The function RiordanSquare is defined in A321620.
Motzkin := (1 - x - sqrt(1 - 2*x - 3*x^2))/(2*x^2); RiordanSquare(Motzkin, 10);
  • Mathematica
    (* The function RiordanSquare is defined in A321620. *)
Motzkin = (1 - x - Sqrt[1 - 2 x - 3 x^2])/(2 x^2);
M = RiordanSquare[Motzkin, 10];
M // Flatten (* Jean-François Alcover, Nov 24 2018 *)
  • Sage
    # uses[riordan_square from A321620]
riordan_square((1 - x - sqrt(1 - 2*x - 3*x^2))/(2*x^2), 10)
Showing 1-3 of 3 results.

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