A365567 -id:A365567 - cp's OEIS frontend

A094417 Generalized ordered Bell numbers Bo(4,n).

1, 4, 36, 484, 8676, 194404, 5227236, 163978084, 5878837476, 237109864804, 10625889182436, 523809809059684, 28168941794178276, 1641079211868751204, 102961115527874385636, 6921180217049667005284, 496267460209336700111076, 37807710659221213027893604

Offset: 0

Ralf Stephan, May 02 2004

nonn

Fourth row of array A094416, which has more information.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.

Cf. A000670, A004123, A032033, A050353, A094418, A094419, A131689.

Cf. A346983, A354242, A365567.

m:=20; R:=LaurentSeriesRing(RationalField(), m); b:=Coefficients(R!(1/(5 - 4*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // Bruno Berselli, Mar 17 2014

a:= proc(n) option remember;
      `if`(n=0, 1, 4* add(binomial(n, k) *a(k), k=0..n-1))
    end:
seq(a(n), n=0..20);

max = 16; f[x_] := 1/(5-4*E^x); CoefficientList[Series[f[x], {x, 0, max}], x]*Range[0, max]! (* Jean-François Alcover, Nov 14 2011, after g.f. *)

my(N=25,x='x+O('x^N)); Vec(serlaplace(1/(5 - 4*exp(x)))) \\ Joerg Arndt, Jan 15 2024

def A094416(n,k): return sum(factorial(j)*n^j*stirling_number2(k,j) for j in range(k+1)) # array
def A094417(k): return A094416(4,k)
[A094417(n) for n in range(31)] # G. C. Greubel, Jan 12 2024

E.g.f.: 1/(5 - 4*exp(x)).
a(n) = 4 * A050353(n) for n>0.
a(n) = Sum_{k=0..n} A131689(n,k) * 4^k. - Philippe Deléham, Nov 03 2008
E.g.f.: A(x) with A_n = 4 * Sum_{k=0..n-1} C(n,k) * A_k; A_0 = 1. - Vladimir Kruchinin, Jan 27 2011
G.f.: 2/G(0), where G(k)= 1 + 1/(1 - 8*x*(k+1)/(8*x*(k+1) - 1 + 10*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
a(n) = log(5/4)*int {x = 0..inf} (floor(x))^n * (5/4)^(-x) dx. - Peter Bala, Feb 14 2015
a(0) = 1; a(n) = 4 * a(n-1) - 5 * Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023
From Seiichi Manyama, Jun 01 2025: (Start)
a(n) = (-1)^(n+1)/5 * Li_{-n}(5/4), where Li_{n}(x) is the polylogarithm function.
a(n) = (1/5) * Sum_{k>=0} k^n * (4/5)^k.
a(n) = (4/5) * Sum_{k=0..n} 5^k * (-1)^(n-k) * A131689(n,k) for n > 0. (End)

A346983 Expansion of e.g.f. 1 / (5 - 4 * exp(x))^(1/4).

Original entry on oeis.org

1, 1, 6, 61, 891, 16996, 400251, 11217781, 364638336, 13486045291, 559192836771, 25691965808026, 1295521405067181, 71131584836353861, 4224255395774155566, 269791923787785076921, 18439806740525320993551, 1342957106015632474616956, 103824389511747541791086511

Offset: 0

Ilya Gutkovskiy, Aug 09 2021

Stirling transform of A007696.

Seiichi Manyama, Table of n, a(n) for n = 0..360

Cf. A000670, A007696, A305404, A346982, A346984, A346985, A352117, A352118, A352119.

Cf. A094417, A354242, A365567.

g:= proc(n) option remember; `if`(n<2, 1, (4*n-3)*g(n-1)) end:
b:= proc(n, m) option remember;
     `if`(n=0, g(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..18);  # Alois P. Heinz, Aug 09 2021

nmax = 18; CoefficientList[Series[1/(5 - 4 Exp[x])^(1/4), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] 4^k Pochhammer[1/4, k], {k, 0, n}], {n, 0, 18}]

a(n) = Sum_{k=0..n} Stirling2(n,k) * A007696(k).
a(n) ~ n! / (Gamma(1/4) * 5^(1/4) * n^(3/4) * log(5/4)^(n + 1/4)). - Vaclav Kotesovec, Aug 14 2021
O.g.f. (conjectural): 1/(1 - x/(1 - 5*x/(1 - 5*x/(1 - 10*x/(1 - 9*x/(1 - 15*x/(1 - ... - (4*n-3)*x/(1 - 5*n*x/(1 - ... ))))))))) - a continued fraction of Stieltjes-type. - Peter Bala, Aug 22 2023
a(0) = 1; a(n) = Sum_{k=1..n} (4 - 3*k/n) * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023
a(0) = 1; a(n) = a(n-1) - 5*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023

A365584 Expansion of e.g.f. 1 / (1 + 4 * log(1-x))^(3/4).

Original entry on oeis.org

1, 3, 24, 300, 5100, 109692, 2854344, 87164088, 3055516800, 120916282368, 5331444120576, 259168711406976, 13769882994784896, 793844510730348672, 49353915922852214016, 3291455140392403401984, 234388011123877880424960, 17750517946502792294592000

Offset: 0

Seiichi Manyama, Sep 10 2023

nonn

Cf. A347016, A354241, A354264.

Cf. A365567.

a[n_] := Sum[Product[4*j + 3, {j, 0, k - 1}] * Abs[StirlingS1[n, k]], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Sep 10 2023 *)

a(n) = sum(k=0, n, prod(j=0, k-1, 4*j+3)*abs(stirling(n, k, 1)));

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (4*j+3)) * |Stirling1(n,k)|.
a(0) = 1; a(n) = Sum_{k=1..n} (4 - k/n) * (k-1)! * binomial(n,k) * a(n-k).
a(n) ~ Gamma(1/4) * n^(n + 1/4) / (2^(3/2) * sqrt(Pi) * (exp(1/4) - 1)^(n + 3/4) * exp(3*n/4)). - Vaclav Kotesovec, Nov 11 2023

cp's OEIS Frontend

A094417 Generalized ordered Bell numbers Bo(4,n).

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A346983 Expansion of e.g.f. 1 / (5 - 4 * exp(x))^(1/4).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A365584 Expansion of e.g.f. 1 / (1 + 4 * log(1-x))^(3/4).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula