A294188 -id:A294188 - cp's OEIS frontend

A294189 E.g.f.: exp(2*(1/(1-x)^2 - 1)).

1, 4, 28, 256, 2848, 37024, 547936, 9064192, 165339904, 3290839552, 70870959616, 1640130678784, 40555349598208, 1066271901822976, 29684252411219968, 871864036043259904, 26931181039027879936, 872418396180001718272, 29564373329109844885504

Offset: 0

Seiichi Manyama, Oct 24 2017

Seiichi Manyama, Table of n, a(n) for n = 0..427

Column k=2 of A294188.

Cf. A136658.

nmax = 20; CoefficientList[Series[E^(2*(1/(1-x)^2 - 1)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 28 2025 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp(2*(1/(1-x)^2-1))))

From Vaclav Kotesovec, Aug 28 2025: (Start)
a(n) = (3*n+1)*a(n-1) - 3*(n-2)*(n-1)*a(n-2) + (n-3)*(n-2)*(n-1)*a(n-3).
a(n) ~ 2^(1/3) * 3^(-1/2) * exp(-4/3 + 2^(1/3)*n^(1/3) + 3*2^(-1/3)*n^(2/3) - n) * n^(n - 1/6) * (1 + 19/(27*2^(1/3)*n^(1/3)) - 11/(3645*2^(2/3)*n^(2/3))). (End)

A294190 Expansion of e.g.f. exp(3*(1/(1-x)^3 - 1)).

Original entry on oeis.org

1, 9, 117, 1881, 35505, 763209, 18309861, 483213033, 13876068609, 429885495945, 14269174133301, 504579048629049, 18917482749919857, 748904109963227721, 31194958415494252005, 1363010131166317306569, 62300347592864208824961

Offset: 0

Seiichi Manyama, Oct 24 2017

Seiichi Manyama, Table of n, a(n) for n = 0..410

Column k=3 of A294188.

Cf. A202826.

With[{nn=20},CoefficientList[Series[Exp[3(1/(1-x)^3-1)],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Mar 03 2018 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp(3*(1/(1-x)^3-1))))

E.g.f.: exp(3*(1/(1-x)^3 - 1)).
From Vaclav Kotesovec, Aug 28 2025: (Start)
a(n) = (4*n+5)*a(n-1) - 6*(n-2)*(n-1)*a(n-2) + 4*(n-3)*(n-2)*(n-1)*a(n-3) - (n-4)*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 3^(1/4) * exp(-9/4 + 5*sqrt(3)*n^(1/4)/8 + 3*sqrt(n)/2 + 4*n^(3/4)/sqrt(3) - n) * n^(n - 1/8)/2 * (1 + 1333*sqrt(3)/(2560*n^(1/4))). (End)

A294192 a(n) = n! * [x^n] exp(n*(1/(1-x)^n- 1)).

Original entry on oeis.org

1, 1, 28, 1881, 241696, 50903725, 15950981376, 6965675824177, 4044321664205824, 3013398202413705561, 2802908316869098624000, 3183461016834811739809321, 4335741846110695855971852288, 6974958097518147761490109983781

Offset: 0

Seiichi Manyama, Oct 24 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..100

Main diagonal of A294188.

Cf. A294047.

S:= series(exp(n*(1/(1-x)^n-1)),x,31):
seq(n!*coeff(S,x,n),n=0..30); # Robert Israel, Oct 24 2017

Table[n! SeriesCoefficient[Exp[n (1 / (1 - x)^n - 1)], {x, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Oct 25 2017 *)

cp's OEIS Frontend

A294189 E.g.f.: exp(2*(1/(1-x)^2 - 1)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A294190 Expansion of e.g.f. exp(3*(1/(1-x)^3 - 1)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A294192 a(n) = n! * [x^n] exp(n*(1/(1-x)^n- 1)).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica