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.

A166337 a(n)=(2n+0^n)*C(4n,2n).

Original entry on oeis.org

1, 12, 280, 5544, 102960, 1847560, 32449872, 561632400, 9617286240, 163352435400, 2756930576400, 46290177201840, 773942488394400, 12893881856650704, 214163336821292320, 3547937446945842720
Offset: 0

Views

Author

Paul Barry, Oct 12 2009

Keywords

Comments

The aerated sequence gives the central coefficients in number triangle A166335.

Links

Programs

  • Mathematica
    Join[{1},Table[(2n)*Binomial[4n,2n],{n,20}]] (* Harvey P. Dale, Jan 21 2015 *)

Formula

D-finite with recurrence (2*n-1)*(n-1)*a(n) -2*(4*n-1)*(4*n-3)*a(n-1)=0. - R. J. Mathar, Feb 27 2023

A166336 Expansion of (1 - 4*x + 7*x^2 - 4*x^3 + x^4)/(1 - 7*x + 17*x^2 - 17*x^3 + 7*x^4 - x^5).

Original entry on oeis.org

1, 3, 11, 39, 131, 421, 1309, 3971, 11823, 34691, 100611, 289033, 823801, 2332419, 6566291, 18394911, 51310979, 142587181, 394905493, 1090444931, 3002921271, 8249479163, 22612505091, 61857842449, 168903452401, 460409998851
Offset: 0

Views

Author

Paul Barry, Oct 12 2009

Keywords

Comments

The diagonal sums of number triangle A166335 are 1, 0, 3, 0, 11, 0, ...

Links

Crossrefs

Cf. A001870, A166335.

Programs

  • Mathematica
    CoefficientList[Series[(1-4x+7x^2-4x^3+x^4)/(1-7x+17x^2-17x^3+7x^4-x^5),{x,0,30}],x] (* or *) LinearRecurrence[{7,-17,17,-7,1},{1,3,11,39,131},30] (* Harvey P. Dale, Jul 05 2014 *)

Formula

G.f.: (1 - 4*x + 7*x^2 - 4*x^3 + x^4)/((1 - x)*(1 - 3*x + x^2)^2);
a(n) = 1 + 2*Sum{k=0..n} k*C(n + k, 2*k) = 1 + 2*Sum{k=0..n} (n-k)*C(2*n - k, k) = 1 + 2*A001870(n).
a(0) = 1, a(1) = 3, a(2) = 11, a(3) = 39, a(4) = 131, and a(n) = -17*a(n-1) + 17*a(n-2) - 7*a(n-3) + a(n-4) for n >= 4. - Harvey P. Dale, Jul 05 2014
Showing 1-2 of 2 results.

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