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.

A116392 Riordan array (1/sqrt(1-2*x-3*x^2), 1/sqrt(1-2*x-3*x^2) -1).

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 7, 13, 7, 1, 19, 42, 32, 10, 1, 51, 131, 128, 60, 13, 1, 141, 406, 475, 292, 97, 16, 1, 393, 1247, 1685, 1267, 561, 143, 19, 1, 1107, 3814, 5800, 5112, 2804, 962, 198, 22, 1, 3139, 11623, 19540, 19624, 12748, 5464, 1522, 262, 25, 1, 8953, 35334
Offset: 0

Views

Author

Paul Barry, Feb 12 2006

Keywords

Comments

Triangle, read by rows, given by [1, 2, -1, -1, 2, 1/2, 1/2, 2, -1, -1, 2, 1/2, 1/2, 2, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 11 2020

Examples

			Triangle begins:
   1;
   1,   1;
   3,   4,   1;
   7,  13,   7,  1;
  19,  42,  32, 10,  1;
  51, 131, 128, 60, 13, 1;

Links

Crossrefs

Row sums are A115967. Diagonal sums are A116394.
Cf. A321620.

Programs

  • Magma
    [[(&+[ Binomial(n,m)*(&+[ (&+[ Round((-1)^(k-j)*4^r* Binomial(k,j)*Binomial(j, m-2*r)*Gamma(r+(j+1)/2)/(Factorial(r)*Gamma((j+1)/2))) : r in [0..Floor(n/2)]]) : j in [0..k]]): m in [0..n]]) : k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 23 2019
  • Maple
    # The function RiordanSquare is defined in A321620.
RiordanSquare(1/sqrt(1 - 2*x - 3*x^2), 10); # Peter Luschny, Feb 15 2020
  • Mathematica
    t[n_,k_]:= Sum[(-1)^(k-j)*Binomial[k,j]*Sum[4^r*Binomial[r+(j-1)/2, r]* Binomial[j, n-2*r], {r,0,Floor[n/2]}], {j,0,k}]; Table[Sum[Binomial[n, j]*t[j,k], {j,0,n}] {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, May 23 2019 *)
  • PARI
    t(n,k) = sum(j=0,k, sum(r=0,floor(n/2), (-1)^(k-j)*4^r* binomial(k,j)*binomial(r+(j-1)/2, r)*binomial(j, n-2*r) ));
T(n,k) = sum(j=0,n, binomial(n,j)*t(j,k)); \\ G. C. Greubel, May 23 2019
  • Sage
    [[sum(binomial(n,m)*sum( sum( (-1)^(k-j)*4^r* binomial(k,j)* binomial(r+(j-1)/2, r)*binomial(j, m-2*r) for r in (0..floor(n/2))) for j in (0..k)) for m in (0..n)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 23 2019

Formula

Number triangle T(n,k) = Sum_{j=0..n} C(n,j)*A116389(j,k).

A116390 Expansion of 1/(2*sqrt(1-4*x^2)-x-1).

Original entry on oeis.org

1, 1, 5, 9, 33, 73, 233, 569, 1693, 4353, 12477, 32985, 92637, 248673, 690549, 1869513, 5158881, 14033161, 38587193, 105246041, 288818305, 788939769, 2162574513, 5912375033, 16196093881, 44300854441, 121311490937
Offset: 0

Views

Author

Paul Barry, Feb 12 2006

Keywords

Comments

Hankel transform is 4^n. - Paul Barry, Jan 19 2011

Links

Crossrefs

Row sums of number triangle A116389.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/(2*Sqrt(1-4*x^2)-x-1) )); // G. C. Greubel, May 23 2019
  • Mathematica
    CoefficientList[Series[1/(2*Sqrt[1-4*x^2]-x-1), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 03 2014 *)
  • PARI
    my(x='x+O('x^30)); Vec(1/(2*sqrt(1-4*x^2)-x-1)) \\ G. C. Greubel, May 23 2019
  • Sage
    (1/(2*sqrt(1-4*x^2)-x-1)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 23 2019

Formula

a(n) = Sum_{k=0..n} Sum_{j=0..k} Sum_{i=0..floor(n/2)} (-1)^(k-j)*C(k,j) *C(i+(j-1)/2,i)*C(j,n-2*i)*4^i.
a(n) = Sum_{k=0..floor((n+1)/2)} (C(n,k) - C(n,k-1))*A006130(n-2*k). - Paul Barry, Jan 19 2011
Starting with offset 1, let M = an infinite tridiagonal matrix with [1,0,0,0,...] in the main diagonal and [2,1,1,1,...] in the super and subdiagonals. Let V = vector [1,0,0,0,...]. The sequence = iterates of M*V as to the leftmost column. - Gary W. Adamson, Jun 08 2011
D-finite with recurrence: -3*n*a(n) + 2*n*a(n-1) + (29*n-36)*a(n-2) + 8*(3-n)*a(n-3) + 68*(3-n)*a(n-4)=0. - R. J. Mathar, Aug 09 2012
a(n) ~ (1+2/sqrt(13)) * (1+2*sqrt(13))^n / 3^(n+1). - Vaclav Kotesovec, Feb 03 2014

A116391 Expansion of 1/((1+x)*(sqrt(1-4*x^2)-x)).

Original entry on oeis.org

1, 0, 3, 2, 11, 14, 47, 78, 217, 408, 1039, 2086, 5065, 10560, 24931, 53194, 123403, 267222, 612903, 1340222, 3050679, 6714946, 15205967, 33622158, 75864835, 168275790, 378743151, 841959974, 1891648931, 4211866694, 9450828951
Offset: 0

Views

Author

Paul Barry, Feb 12 2006

Keywords

Links

Crossrefs

Diagonal sums of A116389.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/((1+x)*(Sqrt(1-4*x^2)-x)) )); // G. C. Greubel, May 23 2019
  • Mathematica
    CoefficientList[Series[1/((1+x)(Sqrt[1-4(x^2) ]-x)),{x,0,40}],x] (* Harvey P. Dale, Sep 25 2018 *)
  • PARI
    my(x='x+O('x^30)); Vec(1/((1+x)*(sqrt(1-4*x^2)-x))) \\ G. C. Greubel, May 23 2019
  • Sage
    (1/((1+x)*(sqrt(1-4*x^2)-x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 23 2019

Formula

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..k} Sum_{i=0..floor((n-k)/2)} (-1)^(k-j)*C(k,j)*C(i+(j-1)/2,i)*C(j,n-k-2i)*4^i.
Conjecture D-finite with recurrence: n*a(n) +(n)*a(n-1) +3*(-3*n+4)*a(n-2) +3*(-3*n+4)*a(n-3) +20*(n-3)*a(n-4) +20*(n-3)*a(n-5)=0. - R. J. Mathar, Jan 23 2020
a(n) ~ 5^(n/2)/(1+sqrt(5)). - Vaclav Kotesovec, Nov 19 2021
Showing 1-3 of 3 results.

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