A202830 -id:A202830 - cp's OEIS frontend

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

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

Offset: 0

Paul D. Hanna, Dec 25 2011

nonn

			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! + ...

G. C. Greubel, Table of n, a(n) for n = 0..350

Cf. A202830, A202827, A202828, A202831, A202833, A202835, A202836.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( Exp(4*x/(1-3*x))/Sqrt(1-9*x^2) ))); // G. C. Greubel, Jun 21 2022

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 *)

{a(n)=n!*polcoeff(exp(4*x/(1-3*x)+x*O(x^n))/sqrt(1-9*x^2+x*O(x^n)),n)}

{a(n)=sum(k=0,n\2,2^(n-3*k)*3^k*n!/((n-2*k)!*k!))^2}

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

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(2x + (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

Andrew Howroyd, Oct 07 2024

			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 ...
     ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
Arvind Ayyer, Hiranya Kishore Dey and Digjoy Paul, How large is the character degree sum compared to the character table sum for a finite group?, arXiv preprint arXiv:2406.06036, [math.RT], 2024.

Columns 0..5 are A000079, A005425, A000898, A202830, A193778, A202832.

Cf. A359762, A373625.

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

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

Ilya Gutkovskiy, Nov 20 2020

nonn

Cf. A000085, A000898, A202830, A294119.

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}]

my(x='x+O('x^30)); Vec(serlaplace(exp((3*x*(2 + x)/2)))) \\ Michel Marcus, Nov 21 2020

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

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

SageMath

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

SageMath

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

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

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