A088312 -id:A088312 - cp's OEIS frontend

A088313 Number of "sets of lists" (cf. A000262) with an odd number of lists.

0, 1, 2, 7, 36, 241, 1950, 18271, 193256, 2270017, 29272410, 410815351, 6231230412, 101560835377, 1769925341366, 32838929702671, 646218442877520, 13441862819232001, 294656673023216946, 6788407001443004647, 163962850573039534580, 4142654439686285737201

Offset: 0

Vladeta Jovovic, Nov 05 2003

From Peter Bala, Mar 27 2022: (Start)
a(2*n) is even; in fact, 2*n*(2*n-1)*(2n-2) divides a(2*n). a(2*n+1) is odd.
For a positive integer k, a(n+2*k) - a(n) is divisible by k. Thus the sequence obtained by taking a(n) modulo k is purely periodic with period 2*k. Calculation suggests that when k is even the exact period equals k, and when k is odd the exact period equals 2*k. (End)

Alois P. Heinz, Table of n, a(n) for n = 0..444
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
N. J. A. Sloane, LAH transform

Cf. A000262, A001710, A027187, A024429, A024430, A088312, A109777, A111884.

R:=PowerSeriesRing(Rationals(), 30); [0] cat Coefficients(R!(Laplace( Sinh(x/(1-x)) ))); // G. C. Greubel, Dec 13 2022

b:= proc(n, t) option remember; `if`(n=0, t, add(
      b(n-j, 1-t)*binomial(n-1, j-1)*j!, j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, May 10 2016
A088313 := n -> ifelse(n=0, 0, n!*hypergeom([1/2 - n/2, 1 - n/2], [1/2, 1, 3/2], 1/4)): seq(simplify(A088313(n)), n = 0..21); # Peter Luschny, Dec 14 2022

With[{m=30}, CoefficientList[Series[Sinh[x/(1-x)], {x,0,m}], x] * Range[0,m]!] (* Vaclav Kotesovec, Jul 04 2015 *)

my(x='x+O('x^66)); concat(0, Vec(serlaplace(sinh(x/(1-x))))) \\ Joerg Arndt, Jul 16 2013

def A088313_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( sinh(x/(1-x)) ).egf_to_ogf().list()
A088313_list(40) # G. C. Greubel, Dec 13 2022

E.g.f.: sinh(x/(1-x)).
a(n) = Sum_{k=1..floor((n+1)/2)} n!/(2*k-1)!*binomial(n-1, 2*k-2).
E.g.f.: sinh(x/(1-x)) = x/(2-2*x)*E(0), where E(k)= 1 + 1/( 1 - x^2/(x^2 + 2*(1-x)^2*(k+1)*(2*k+3)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 16 2013
a(n) ~ 2^(-3/2) * n^(n-1/4) * exp(2*sqrt(n)-n-1/2). - Vaclav Kotesovec, Jul 04 2015
a(n) = (1/2)*(A000262(n) - (-1)^n*A111884(n)). - Peter Bala, Mar 27 2022
a(n) = n!*hypergeom([1/2 - n/2, 1 - n/2], [1/2, 1, 3/2], 1/4) for n >= 1. - Peter Luschny, Dec 14 2022

a(0)=0 prepended by Alois P. Heinz, May 10 2016

A317409 Expansion of e.g.f. cos(x/(1 - x)).

Original entry on oeis.org

1, 0, -1, -6, -35, -220, -1501, -10962, -83495, -632952, -4260601, -13852190, 355180981, 12991115436, 320077652075, 7153866992790, 155785273182001, 3395838000334352, 75000970329466895, 1687941779356532682, 38803334491247820301, 911633573138881234740, 21870615120012355726259

Offset: 0

Ilya Gutkovskiy, Jul 27 2018

sign

Lah transform of the sequence 1, 0, -1, 0, 1, 0, -1, 0, ...

Robert Israel, Table of n, a(n) for n = 0..446
N. J. A. Sloane, Transforms

Cf. A056594, A088312, A219613, A317406, A317410.

a:=series(cos(x/(1 - x)), x=0, 22): seq(n!*coeff(a, x, n), n=0..21); # Paolo P. Lava, Mar 26 2019

nmax = 22; CoefficientList[Series[Cos[x/(1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^k Binomial[n - 1, 2 k - 1] n!/(2 k)!, {k, 0, Floor[n/2]}], {n, 0, 22}]
Join[{1}, Table[(1 - n) n! HypergeometricPFQ[{1 - n/2, 3/2 - n/2}, {3/2, 3/2, 2}, -1/4]/2, {n, 22}]]

my(x='x + O('x^25)); Vec(serlaplace(cos(x/(1 - x)))) \\ Michel Marcus, Mar 26 2019

a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-1,2*k-1)*n!/(2*k)!.

-2*(2*n + 3)*(n + 2)*(n + 1)*a(n + 1) + (6*n^2 + 24*n + 25)*a(n + 2) - 2*(2*n + 5)*a(n + 3) + a(n + 4) + n*(n + 2)*(n + 1)^2*a(n)=0. - Robert Israel, Mar 26 2019

A014297 a(n) = n! * C(n+2, 2) * 2^(n+1).

Original entry on oeis.org

2, 12, 96, 960, 11520, 161280, 2580480, 46448640, 928972800, 20437401600, 490497638400, 12752938598400, 357082280755200, 10712468422656000, 342798989524992000, 11655165643849728000, 419585963178590208000, 15944266600786427904000, 637770664031457116160000

Offset: 0

Emeric Deutsch

Partition the set {1,2,...,n+2} into an even number of subsets. Arrange (linearly order) the elements within each subset and then arrange the subsets. - Geoffrey Critzer, Mar 03 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 506
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.

Essentially the same as A052564.

Cf. A088312.

List([0..20], n-> 2^n*Factorial(n+2)); # G. C. Greubel, May 05 2019

[2^n*Factorial(n+2): n in [0..20]]; // G. C. Greubel, May 05 2019

seq(count(Permutation(n+1))*count(Composition(n)),n=1..17); # Zerinvary Lajos, Oct 16 2006

Drop[CoefficientList[Series[(1-x)^2/(1-2x), {x, 0, 20}], x]* Table[n!, {n, 0, 20}], 2] (* Geoffrey Critzer, Mar 03 2010 *)
Part[#, Range[1, Length[#], 1]]&@(Array[#!&, Length[#], 0]*#)&@CoefficientList[Series[2/(1 - 2*x)^3, {x, 0, 20}], x]// ExpandAll (* Vincenzo Librandi, Jan 04 2013 - after Olivier Gérard in A213068 *)
Table[n!Binomial[n+2,2]2^(n+1),{n,0,30}] (* Harvey P. Dale, Dec 27 2022 *)

a(n) = (n+2)!*2^n; \\ Joerg Arndt, May 05 2019

[2^n*factorial(n+2) for n in (0..20)] # G. C. Greubel, May 05 2019

a(n) = Sum_{k=0..n} (n+2)!*C(n,k).
Prepend the sequence with 1,0, then e.g.f. is (1-x)^2/(1-2*x). - Geoffrey Critzer, Mar 03 2010
E.g.f.: 2/(1-2*x)^3. - R. J. Mathar, Nov 27 2011
a(n) = 2*A051578(n). - R. J. Mathar, Apr 26 2017
a(n) = (n+2)! * 2^n. - Joerg Arndt, May 05 2019
From Amiram Eldar, Jul 04 2020: (Start)
Sum_{n>=0} 1/a(n) = 4*sqrt(e) - 6.
Sum_{n>=0} (-1)^n/a(n) = 4/sqrt(e) - 2. (End)

A386474 Number of sets of lists of [n] such that no list is longer than than the total number of lists.

Original entry on oeis.org

1, 1, 1, 7, 25, 141, 1171, 9913, 85233, 907273, 11010691, 143824341, 1988010553, 29605763773, 475664908083, 8284952367721, 153508912353121, 2997209814190353, 61485486404453443, 1326994255131585373, 30144049509450774441, 718905298680190094341, 17940822818538396541843

Offset: 0

John Tyler Rascoe, Jul 23 2025

Here sets of lists are set partitions of [n] such that the elements within each block are ordered but the blocks themselves are unordered.

			a(3) = 7 counts: {(1),(2),(3)}, {(1),(2,3)}, {(1),(3,2)}, {(1,2),(3)}, {(1,3),(2)}, {(2),(3,1)}, {(2,1),(3)}.

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A000262, A077229, A088009, A088311, A088312, A088313, A113775, A351884.

b:= proc(n, m, l) option remember; `if`(m>n+l, 0, `if`(n=0, 1,
      add(b(n-j, max(m, j), l+1)*(n-1)!*j/(n-j)!, j=1..n)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..22);  # Alois P. Heinz, Jul 23 2025

With[{m = 22}, CoefficientList[1 + Series[Sum[((x - x^(i + 1))/(1 - x))^i/i!, {i, 1, m}], {x, 0, m}], x] * Range[0, m]!] (* Amiram Eldar, Jul 24 2025 *)

R_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(sum(i=0,N,((x-x^(i+1))/(1-x))^i/i!)))}

E.g.f.: Sum_{i>=0} ((x - x^(i+1))/(1 - x))^i / i!.

A386497 Number of sets of lists of [n] such that one list is the largest.

Original entry on oeis.org

1, 1, 2, 12, 60, 440, 3390, 33852, 338072, 4116240, 51776730, 736751180, 11075784852, 183142075272, 3157190863190, 59336602681020, 1164223828582320, 24348331444705952, 533422896546272562, 12365952739192923660, 298208300418298756460, 7570420981014167756760

Offset: 0

John Tyler Rascoe, Jul 23 2025

Here sets of lists are set partitions of [n] such that the elements within each block are ordered but the blocks themselves are unordered.

			a(3) = 12 counts: {(1),(2,3)}, {(1),(3,2)}, {(1,2),(3)}, {(1,3),(2)}, {(2),(3,1)}, {(2,1),(3)}, {(1,2,3)}, {(1,3,2)}, {(2,1,3)}, {(2,3,1)}, {(3,1,2)}, {(3,2,1)}.

Alois P. Heinz, Table of n, a(n) for n = 0..444

Cf. A000262, A088009, A088311, A088312, A088313, A097979, A113775, A351884, A386474.

b:= proc(n, m, t) option remember; `if`(n=0, t, add(b(n-j, max(m, j),
      `if`(j>m, 1, `if`(j=m, 0, t)))*(n-1)!*j/(n-j)!, j=1..n))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..21);  # Alois P. Heinz, Jul 23 2025

With[{m = 21}, CoefficientList[Series[1 + Sum[x^j*Exp[(x - x^j)/(1 - x)], {j, 1, m}], {x, 0, m}], x] * Range[0, m]!] (* Amiram Eldar, Jul 24 2025 *)

B_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(1+sum(j=1,N, x^j*exp((x-x^j)/(1-x)))))}

E.g.f.: 1 + Sum_{j>0} x^j * exp((x - x^j)/(1 - x)).

cp's OEIS Frontend

A088313 Number of "sets of lists" (cf. A000262) with an odd number of lists.

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A317409 Expansion of e.g.f. cos(x/(1 - x)).

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A014297 a(n) = n! * C(n+2, 2) * 2^(n+1).

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A386474 Number of sets of lists of [n] such that no list is longer than than the total number of lists.

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A386497 Number of sets of lists of [n] such that one list is the largest.

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Maple

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