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.

A202829 Expansion of e.g.f.: exp(4*x/(1-3*x)) / sqrt(1-9*x^2).

Original entry on oeis.org

1, 4, 49, 676, 13225, 293764, 7890481, 236359876, 8052729169, 300797402500, 12388985000401, 551925653637604, 26614517015830969, 1373655853915667716, 75803216516463190225, 4440662493517062816004, 275697752917311709134241, 18052104090118575573856516
Offset: 0

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Author

Paul D. Hanna, Dec 25 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + 4*x + 49*x^2/2! + 676*x^3/3! + 13225*x^4/4! + 293764*x^5/5! + ...
were A(x) = 1 + 2^2*x + 7^2*x^2/2! + 26^2*x^3/3! + 115^2*x^4/4! + 542^2*x^5/5! + ... + A202830(n)^2*x^n/n! + ...
		

Crossrefs

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( Exp(4*x/(1-3*x))/Sqrt(1-9*x^2) ))); // G. C. Greubel, Jun 21 2022
    
  • Mathematica
    With[{nn=20},CoefficientList[Series[Exp[(4x)/(1-3x)]/Sqrt[1-9x^2],{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Mar 09 2012 *)
  • PARI
    {a(n)=n!*polcoeff(exp(4*x/(1-3*x)+x*O(x^n))/sqrt(1-9*x^2+x*O(x^n)),n)}
    
  • PARI
    {a(n)=sum(k=0,n\2,2^(n-3*k)*3^k*n!/((n-2*k)!*k!))^2}
    
  • SageMath
    def A202829_list(prec):
        P. = PowerSeriesRing(QQ, prec)
        return P( exp(4*x/(1-3*x))/sqrt(1-9*x^2) ).egf_to_ogf().list()
    A202829_list(40) # G. C. Greubel, Jun 21 2022

Formula

a(n) = A202830(n)^2, where the e.g.f. of A202830 is exp(2*x + 3*x^2/2).
a(n) = ( Sum_{k=0..[n/2]} 2^(n-3*k)*3^k * n!/((n-2*k)!*k!) )^2.
a(n) ~ n^n*exp(4*sqrt(n/3)-2/3-n)*3^n/2. - Vaclav Kotesovec, May 23 2013
D-finite with recurrence: a(n) = (3*n+1)*a(n-1) + 3*(n-1)*(3*n+1)*a(n-2) - 27*(n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013

A376826 Array read by antidiagonals: T(n,k) = n! * [x^n] exp(2*x + (k/2)*x^2), n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 2, 5, 8, 1, 2, 6, 14, 16, 1, 2, 7, 20, 43, 32, 1, 2, 8, 26, 76, 142, 64, 1, 2, 9, 32, 115, 312, 499, 128, 1, 2, 10, 38, 160, 542, 1384, 1850, 256, 1, 2, 11, 44, 211, 832, 2809, 6512, 7193, 512, 1, 2, 12, 50, 268, 1182, 4864, 15374, 32400, 29186, 1024
Offset: 0

Views

Author

Andrew Howroyd, Oct 07 2024

Keywords

Examples

			Array begins:
======================================================
n\k |   0    1    2     3     4     5     6      7 ...
----+-------------------------------------------------
  0 |   1    1    1     1     1     1     1      1 ...
  1 |   2    2    2     2     2     2     2      2 ...
  2 |   4    5    6     7     8     9    10     11 ...
  3 |   8   14   20    26    32    38    44     50 ...
  4 |  16   43   76   115   160   211   268    331 ...
  5 |  32  142  312   542   832  1182  1592   2062 ...
  6 |  64  499 1384  2809  4864  7639 11224  15709 ...
  7 | 128 1850 6512 15374 29696 50738 79760 118022 ...
     ...
		

Crossrefs

Programs

  • PARI
    T(n,k) = {sum(i=0, n\2, binomial(n,2*i) * 2^(n-2*i) * k^i * (2*i)!/(2^i*i!))}

Formula

E.g.f. of column k: exp(2*x + k*x^2/2).
Column k is the binomial transform of column k of A359762.
T(n,k) = Sum_{i=0..floor(n/2)} binomial(n,2*i) * 2^(n-2*i) * k^i * (2*i-1)!!.
T(n,k) = Sum_{i=0..floor(n/2)} 2^(n-3*i) * k^i * n! / ((n-2*i)! * i!).

A335819 E.g.f.: exp((3/2) * x * (2 + x)).

Original entry on oeis.org

1, 3, 12, 54, 270, 1458, 8424, 51516, 331452, 2230740, 15641424, 113846472, 857706408, 6671592216, 53465326560, 440602852752, 3727748253456, 32332181692464, 287111706003648, 2607272929404000, 24187186030419936, 228997933855499808, 2210786521482955392, 21746223198911853504
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 20 2020

Keywords

Crossrefs

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[(3/2) x (2 + x)], {x, 0, nmax}], x] Range[0, nmax]!
    a[0] = 1; a[1] = 3; a[n_] := a[n] = 3 (a[n - 1] + (n - 1) a[n - 2]); Table[a[n], {n, 0, 23}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp((3*x*(2 + x)/2)))) \\ Michel Marcus, Nov 21 2020

Formula

G.f.: 1 / (1 - 3*x - 3*x^2 / (1 - 3*x - 6*x^2 / (1 - 3*x - 9*x^2 / (1 - 3*x - 12*x^2 / (1 - ...))))), a continued fraction.
D-finite with recurrence a(n) = 3 * (a(n-1) + (n-1) * a(n-2)).
a(n) = Sum_{k=0..n} binomial(n,k) * A000085(k) * A000898(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * A202830(k).
a(n) ~ 3^(n/2) * exp(-3/4 + sqrt(3*n) - n/2) * n^(n/2) / sqrt(2). - Vaclav Kotesovec, Aug 09 2021
Showing 1-3 of 3 results.