A070071 -id:A070071 - cp's OEIS frontend

A204618 a(n) = n^2 * B(n) where B(n) are the Bell numbers, A000110.

0, 1, 8, 45, 240, 1300, 7308, 42973, 264960, 1712907, 11597500, 82106970, 606757968, 4671909853, 37416267112, 311165672625, 2682916389632, 23947947373356, 220992885195516, 2105619936025577, 20689663294148800, 209417588925127191, 2181250417408504332

Offset: 0

Geoffrey Critzer, Jan 17 2012

nonn

Muniru A Asiru, Table of n, a(n) for n = 0..550

Cf. A000110, A070071.

List([0..22],n->n^2*Bell(n)); # Muniru A Asiru, Apr 20 2019

nn=20;a=Exp[Exp[x]-1];Range[0,nn]!CoefficientList[Series[x D[x D[a,x],x],{x,0,nn}],x]
Table[n^2 BellB[n],{n,0,30}] (* Harvey P. Dale, Jul 01 2022 *)

from itertools import count, accumulate, islice
def A204618_gen(): # generator of terms
    yield 0
    blist, b = (1,), 1
    for n in count(1):
        blist = list(accumulate(blist, initial=(b:=blist[-1])))
        yield b*n**2
A204618_list = list(islice(A204618_gen(),20)) # Chai Wah Wu, Jun 22 2022

E.g.f.: (x+x^2+x^2 exp(x))exp(exp(x)+x-1) which is x*A'(x) where A(x) is the e.g.f. for A070071.

A306948 Expansion of e.g.f. (1 + x)log(1 + x)exp(x).

Original entry on oeis.org

0, 1, 3, 5, 8, 9, 19, -15, 216, -1407, 11803, -108483, 1106192, -12363703, 150381243, -1977666743, 27965386320, -423158076351, 6822782712723, -116781368777867, 2114916140765496, -40404117909336247, 812091479233464131, -17130720178674680031, 378423227774537955688

Offset: 0

Ilya Gutkovskiy, Mar 17 2019

sign

Cf. A000110, A002741, A048994, A070071, A216313.

a:=series((1 + x)*log(1 + x)*exp(x),x=0,25): seq(n!*coeff(a, x, n), n=0..24); # Paolo P. Lava, Mar 26 2019

nmax = 24; CoefficientList[Series[(1 + x) Log[1 + x] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] BellB[k] k, {k, 0, n}], {n, 0, 24}]
Table[Sum[(-1)^(k - 1) Binomial[n, k] (n - k + 1) (k - 1)!, {k, 1, n}], {n, 0, 24}]

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000110(k)*k.
a(n) = Sum_{k=1..n} (-1)^(k-1)*binomial(n,k)*(n - k + 1)*(k - 1)!.
a(n) ~ exp(-1) * (-1)^n * n! / n^2. - Vaclav Kotesovec, Mar 18 2019
Conjecture: D-finite with recurrence a(n) +(n-5)*a(n-1) +(-3*n+10)*a(n-2) +3*(n-3)*a(n-3) +(-n+3)*a(n-4)=0. - R. J. Mathar, Aug 20 2021

cp's OEIS Frontend

A204618 a(n) = n^2 * B(n) where B(n) are the Bell numbers, A000110.

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A306948 Expansion of e.g.f. (1 + x)log(1 + x)exp(x).

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cp's OEIS Frontend

A204618 a(n) = n^2 * B(n) where B(n) are the Bell numbers, A000110.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Mathematica

Python

Formula

A306948 Expansion of e.g.f. (1 + x)*log(1 + x)*exp(x).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

A306948 Expansion of e.g.f. (1 + x)log(1 + x)exp(x).