A088313 -id:A088313 - cp's OEIS frontend

A109777 G.f. = f(x), where f(x)^2 = o.g.f. for A088313 (with offset 0).

1, 1, 3, 15, 101, 829, 7891, 84735, 1009065, 13170841, 186798003, 2859068831, 46960097413, 823787983021, 15370572776091, 303929827526887, 6348320745774993, 139663855708967665, 3227812335094695171, 78180132507785056399, 1980181972528939129861, 52344600987011191983613

Offset: 0

N. J. A. Sloane and Nadia Heninger, Aug 15 2005

nonn

			The present sequence has g.f. f(x) = 1 + x + 3*x^2 + 15*x^3 + ...
A088313 [1,2,7,36,242,...] has e.g.f. = sinh(x/(1-x)) = x + x^2 + 7/6*x^3 + 3/2*x^4 + 241/120*x^5 + 65/24*x^6 + 18271/5040*x^7 + ... and (with offset 0) o.g.f. = 1 + 2*x^2 + 7*x^3 + 36*x^4 + ... = f(x)^2.

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

nmax = 22;
f[x_] = Sqrt[Sum[SeriesCoefficient[Sinh[x/(1-x)], {x, 0, n}] n! x^n, {n, 0, nmax}]] + O[x]^nmax // Normal;
List @@ f[x] /. x -> 1 (* Jean-François Alcover, Oct 08 2018 *)

A088312 Number of sets of lists (cf. A000262) with even number of lists.

Original entry on oeis.org

1, 0, 1, 6, 37, 260, 2101, 19362, 201097, 2326536, 29668681, 413257790, 6238931821, 101415565836, 1765092183037, 32734873484250, 644215775792401, 13404753632014352, 293976795292186897, 6775966692145553526, 163735077313046119861, 4138498600079573989140

Offset: 0

Vladeta Jovovic, Nov 05 2003

From Peter Bala, Mar 27 2022: (Start)
a(2*n) is odd ; a(2*n+1) is even.
If k is odd then k*(k-1) divides a(k). Consequently, 6 divides a(6*n+3), 10 divides a(10*n+5), 14 divides a(14*n+7), and in general, if k is odd then 2*k divides a(2*k*n + k).
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
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k
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, A088313, A111884.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Cosh(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, 1):
seq(a(n), n=0..30);  # Alois P. Heinz, May 10 2016
A088312 := n -> ifelse(n=0, 1, (1/2)*(n - 1)*n!*hypergeom([1 - n/2, 3/2 - n/2], [3/2, 3/2, 2], 1/4)): seq(simplify(A088312(n)), n = 0..21); # Peter Luschny, Dec 14 2022

With[{m=30}, CoefficientList[Series[Cosh[x/(1-x)], {x,0,m}], x] * Range[0,m]!] (* Vaclav Kotesovec, Jul 04 2015 *)
Table[Sum[n!/(2*k)! Binomial[n - 1, 2*k - 1], {k, 0, Floor[n/2]}], {n, 0, 30}] (* Emanuele Munarini, Sep 03 2017 *)

def A088312_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( cosh(x/(1-x)) ).egf_to_ogf().list()
A088312_list(40) # G. C. Greubel, Dec 13 2022

E.g.f.: cosh(x/(1-x)).
a(n) = Sum_{k=1..floor(n/2)} n!/(2*k)!*binomial(n-1,2*k-1).
a(n) ~ 2^(-3/2) * n^(n-1/4) * exp(2*sqrt(n)-n-1/2). - Vaclav Kotesovec, Jul 04 2015
a(n+4) - 2*(2*n+5)*a(n+3) + (6*n^2+24*n+23)*a(n+2) - 2*(n+1)*(n+2)*(2*n+3)*a(n+1) + n*(n+1)^2*(n+2)*a(n) = 0. - Emanuele Munarini, Sep 03 2017
a(n) = (1/2)*(A000262(n) + (-1)^n*A111884(n)). - Peter Bala, Mar 27 2022
a(n) = (1/2)*(n-1)*n!*hypergeom([1 - n/2, 3/2 - n/2], [3/2, 3/2, 2], 1/4) for n >= 1. - Peter Luschny, Dec 14 2022

More terms from Vaclav Kotesovec, Jul 04 2015

a(0)-a(1) prepended by Alois P. Heinz, May 10 2016

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!.

A317406 Expansion of e.g.f. sin(x/(1 - x)).

Original entry on oeis.org

0, 1, 2, 5, 12, 1, -450, -6931, -89096, -1120895, -14394150, -191263051, -2638282812, -37716883775, -556075744042, -8385570334051, -127637336779920, -1916072623603199, -27033275598036174, -311878728377256475, -918069644450841860, 120594465496571606401, 6362190374664242284782

Offset: 0

Ilya Gutkovskiy, Jul 27 2018

sign

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

N. J. A. Sloane, Transforms

Cf. A088313, A219613, A317409.

a:=series(sin(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[Sin[x/(1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(k + 1) Binomial[n - 1, 2 k - 2] n!/(2 k - 1)!, {k, Floor[(n + 1)/2]}], {n, 0, 22}]
Join[{0}, Table[n! HypergeometricPFQ[{1/2 - n/2, 1 - n/2}, {1/2, 1, 3/2}, -1/4], {n, 22}]]

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

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

A109777 G.f. = f(x), where f(x)^2 = o.g.f. for A088313 (with offset 0).

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A088312 Number of sets of lists (cf. A000262) with even number of lists.

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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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A317406 Expansion of e.g.f. sin(x/(1 - x)).

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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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