A215916 -id:A215916 - cp's OEIS frontend

A052801 A simple grammar: labeled pairs of sequences of cycles.

1, 2, 8, 46, 342, 3108, 33324, 411360, 5741856, 89379120, 1534623936, 28804923024, 586686138384, 12885385945248, 303537419684064, 7633673997722496, 204125888803996800, 5782960189212871680

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Robert Israel, Table of n, a(n) for n = 0..417
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 759.

Cf. A215916.

Cf. A007840, A354122, A354123.

spec := [S,{C=Cycle(Z),B=Sequence(C),S=Prod(B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

CoefficientList[Series[1/(1+Log[1-x])^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)

makelist(sum((-1)^(n-k)*stirling1(n, k)*(k+1)!, k, 0, n), n, 0, 17); /* Bruno Berselli, May 25 2011 */

E.g.f.: 1/(-1+log(-1/(-1+x)))^2.
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling1(n, k)*(k+1)!. - Vladeta Jovovic, Sep 21 2003
a(n) = D^n(1/(1-x)^2) evaluated at x = 0, where D is the operator exp(x)*d/dx. Cf. A052811. - Peter Bala, Nov 25 2011
a(n) ~ n! * n*exp(n)/(exp(1)-1)^(n+2). - Vaclav Kotesovec, Sep 30 2013
From Anton Zakharov, Aug 07 2016: (Start)
a(n) = A007840(n) + A215916(n).
a(n) = Sum_{k=2..n+1} k!*s(n,k) where s(n,k) is the unsigned Stirling number of the first kind, (A132393). (End)
a(0) = 1; a(n) = Sum_{k=1..n} (k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023

A320344 Expansion of e.g.f. log(1 + x)/(1 - log(1 + x))^2.

Original entry on oeis.org

0, 1, 3, 8, 26, 94, 406, 1896, 10440, 59472, 405264, 2673648, 22396128, 160828368, 1704287568, 11993279232, 177349981824, 957018589056, 25766036316288, 33555346603776, 5403108443855616, -28811285794990080, 1643455634670489600, -21001090458387594240, 692074413969784289280

Offset: 0

Ilya Gutkovskiy, Jan 22 2019

sign

Seiichi Manyama, Table of n, a(n) for n = 0..451

Cf. A001563, A048994, A069321, A215916, A317280, A351280.

seq(n!*coeff(series(log(1+x)/(1-log(1+x))^2,x=0,25),x,n),n=0..24); # Paolo P. Lava, Jan 29 2019

nmax = 24; CoefficientList[Series[Log[1 + x]/(1 - Log[1 + x])^2, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k k!, {k, 0, n}], {n, 0, 24}]

my(N=40, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=0, N, k*log(1+x)^k)))) \\ Seiichi Manyama, Apr 22 2022

a(n) = Sum_{k=0..n} Stirling1(n,k)*A001563(k).

E.g.f.: Sum_{k>=0} k * log(1+x)^k. - Seiichi Manyama, Apr 22 2022

A331798 E.g.f.: -log(1 - x) / ((1 - x) * (1 + log(1 - x))).

Original entry on oeis.org

0, 1, 5, 29, 204, 1714, 16862, 190826, 2447512, 35136696, 558727872, 9754239648, 185546362416, 3820734689472, 84687887312688, 2010622152615504, 50908186083448320, 1369376758488222336, 38998680958184088960, 1172297572938013827456, 37092793335394301708544

Offset: 0

Ilya Gutkovskiy, Jan 26 2020

nonn

Cf. A000254, A001008, A002805, A003713, A007526, A007840, A215916, A331797.

nmax = 20; CoefficientList[Series[-Log[1 - x]/((1 - x) (1 + Log[1 - x])), {x, 0, nmax}], x] Range[0, nmax]!
A007526[n_] := n! Sum[1/k!, {k, 0, n - 1}]; a[n_] := Sum[Abs[StirlingS1[n, k]] A007526[k], {k, 0, n}]; Table[a[n], {n, 0, 20}]
A007840[n_] := Sum[Abs[StirlingS1[n, k]] k!, {k, 0, n}]; a[n_] := Sum[Binomial[n, k] k! HarmonicNumber[k] A007840[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

a(n) = Sum_{k=0..n} = |Stirling1(n,k)| * A007526(k).
a(n) = Sum_{k=1..n} binomial(n,k) * k! * H(k) * A007840(n-k), where H(k) is the k-th harmonic number.
a(n) ~ n! / (1 - exp(-1))^(n+1). - Vaclav Kotesovec, Jan 26 2020

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A052801 A simple grammar: labeled pairs of sequences of cycles.

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A320344 Expansion of e.g.f. log(1 + x)/(1 - log(1 + x))^2.

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A331798 E.g.f.: -log(1 - x) / ((1 - x) * (1 + log(1 - x))).

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