A111884 -id:A111884 - 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

A383992 Series expansion of the exponential generating function exp(arbustive(x)) - 1 where arbustive(x) = (log(1+x) - x^2) / (1+x).

Original entry on oeis.org

0, 1, -4, 3, 40, -330, 1626, -3150, -54592, 1060920, -13022280, 127171440, -889086648, -283184616, 179750627616, -4895777544840, 99124001788800, -1721513264431680, 25736021675994816, -292896125040673728, 639149345262276480, 106178474282318726400

Offset: 0

Michael De Vlieger, May 16 2025

Michael De Vlieger, Table of n, a(n) for n = 0..448
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 32, Table 3, operad "Arbustive".

Cf. A003725, A097388, A111884, A112242, A114285, A177885, A318215, A383990, A383991, A383993, A383994, A383995.

nn = 21; f[x_] := Exp[x] - 1;
Range[0, nn]! * CoefficientList[Series[f[(Log[1 + x] - x^2)/(1 + x)], {x, 0, nn}], x]

A383993 Series expansion of the exponential generating function exp(tridup^!(x)) - 1 where tridup^!(x) = x / ((1+x) * (1+2*x)).

Original entry on oeis.org

0, 1, -5, 25, -119, 301, 5611, -171275, 3574705, -68597639, 1282415131, -23479249199, 409082338105, -6146707844315, 46462772999371, 2072826643602541, -160983324879816479, 8004468391727017585, -352443295329194182085, 14817357881274444545161

Offset: 0

Michael De Vlieger, May 16 2025

The series tridup^!(x) is the inverse for the substitution of the series tridup(x) (given by A001003), given by the suspension of the Koszul dual of tridup. - Bérénice Delcroix-Oger, May 28 2025

Michael De Vlieger, Table of n, a(n) for n = 0..404
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 32, Table 3, triduplicial operad "TriDup".

Cf. A002050, A003725, A097388, A111884, A112242, A177885, A318215, A383990, A383991, A383992, A383994, A383995.

nn = 19; f[x_] := Exp[x] - 1;
Range[0, nn]! * CoefficientList[Series[f[x/((1 + x)*(1 + 2*x))], {x, 0, nn}], x]

A383994 Series expansion of the exponential generating function exp(wnp^!(x)) - 1 where wnp^!(x) = log(1+x) - x^2/(1+x).

Original entry on oeis.org

0, 1, -2, 0, 12, -60, 240, -840, 1680, 15120, -332640, 4656960, -59209920, 735134400, -9098369280, 112345833600, -1365274310400, 15746578848000, -155630893017600, 762963647846400, 22567767443020800, -1126188650069683200, 35900904478389350400

Offset: 0

Michael De Vlieger, May 16 2025

sign

The series wnp^!(x) is the inverse for the substitution of the series wnp(x) (corresponding to A048172), given by the suspension of the Koszul dual of the WithoutNPosets operad. - Bérénice Delcroix-Oger, May 28 2025

Michael De Vlieger, Table of n, a(n) for n = 0..451
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 32, Table 3, operad without n posets "WNP".

Cf. A003725, A084099, A097388, A111884, A112242, A177885, A318215, A383990, A383991, A383992, A383993, A383995.

nn = 22; f[x_] := Exp[x] - 1;
Range[0, nn]! * CoefficientList[Series[f[Log[1 + x] - x^2/(1 + x)], {x, 0, nn}], x]

A383990 Series expansion of the exponential generating function exp(-dend(-x))-1 where dend(x) = (1 - sqrt(1+4x)) / (2x) + 1 (given by A000108).

Original entry on oeis.org

0, 1, -3, 19, -191, 2661, -47579, 1040047, -26888511, 802727209, -27178685459, 1029077910411, -43086906080063, 1976633329627789, -98597207392040811, 5313105048925173991, -307587436319162110079, 19038773384213189214417, -1254686724727364725716131

Offset: 0

Michael De Vlieger, May 16 2025

sign

The series -dend(-x) is the inverse for the substitution of the series dias(x), given by the suspension of the Koszul dual of dias. - Bérénice Delcroix-Oger, May 28 2025

Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 32, Table 3, diassociative operad "Dias".

Cf. A003725, A006531, A097388, A111884, A112242, A177885, A318215, A383991, A383992, A383993, A383994, A383995. Composition of exp(x)-1 with -A000108(-x).

A293133 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^(k+1)/(1+x)).

Original entry on oeis.org

1, 1, 1, 1, 0, -1, 1, 0, 2, 1, 1, 0, 0, -6, 1, 1, 0, 0, 6, 36, -19, 1, 0, 0, 0, -24, -240, 151, 1, 0, 0, 0, 24, 120, 1920, -1091, 1, 0, 0, 0, 0, -120, -360, -17640, 7841, 1, 0, 0, 0, 0, 120, 720, 0, 183120, -56519, 1, 0, 0, 0, 0, 0, -720, -5040, 20160, -2116800

Offset: 0

Seiichi Manyama, Sep 30 2017

			Square array begins:
     1,    1,   1,    1, ...
     1,    0,   0,    0, ...
    -1,    2,   0,    0, ...
     1,   -6,   6,    0, ...
     1,   36, -24,   24, ...
   -19, -240, 120, -120, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..2 give A111884, A293120, A293121.
Rows n=0..1 give A000012, A000007.
Main diagonal gives A000007.
A(n,n-1) gives A000142(n).
Cf. A293053, A293119, A293134,

def f(n)
  return 1 if n < 2
  (1..n).inject(:*)
end
def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
  ary = [1]
  (1..n).each{|i| ary << (-1) ** (k % 2) * (k..i - 1).inject(0){|s, j| s + (-1) ** (j % 2) * f(j + 1) * ncr(i - 1, j) * ary[i - 1 - j]}}
  ary
end
def A293133(n)
  a = []
  (0..n).each{|i| a << A(i, n - i)}
  ary = []
  (0..n).each{|i|
    (0..i).each{|j|
      ary << a[i - j][j]
    }
  }
  ary
end
p A293133(20)

A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = (-1)^k * Sum_{i=k..n-1} (-1)^i*(i+1)!*binomial(n-1,i)*A(n-1-i,k) for n > k.

A293571 E.g.f.: exp(x/(1 + x + x^2)).

Original entry on oeis.org

1, 1, -1, -5, 25, 41, -1049, 2899, 54545, -610415, -1363409, 92652011, -651996311, -10663181255, 262674487895, -529402905149, -68312606260319, 1136414207246369, 7701376416944095, -584076369474366245, 6461047290787787321, 173442620419212050761

Offset: 0

Seiichi Manyama, Oct 12 2017

sign

Robert Israel, Table of n, a(n) for n = 0..446
Robert Israel, Plot of a(n)/(n! exp(sqrt(n))) for n = 2 .. 2500

Cf. A111884, A293572, A293573.

f:= gfun:-rectoproc({a(n) = (3-2*n)*a(n-1) - 3*(n-2)*(n-1)*a(n-2) + (5-2*n)*(n-1)*(n-2)*a(n-3)- (n-4)*(n-3)*(n-2)*(n-1)*a(n-4),a(0)=1,a(1)=1,a(2)=-1,a(3)=-5},a(n),remember):
map(f, [$0..30]); # Robert Israel, Jul 27 2020

N=66; x='x+O('x^N); Vec(serlaplace(exp(x/(1+x+x^2))))

N=66; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, exp(x^(3*k-2)-x^(3*k-1)))))

E.g.f.: Product_{k>0} exp(x^(3*k-2)) / exp(x^(3*k-1)).

a(n) = (3-2*n)*a(n-1) - 3*(n-2)*(n-1)*a(n-2) + (5-2*n)*(n-1)*(n-2)*a(n-3) - (n-4)*(n-3)*(n-2)*(n-1)*a(n-4). - Robert Israel, Jul 27 2020

A293572 E.g.f.: exp(x/(1 + x + x^2 + x^3)).

Original entry on oeis.org

1, 1, -1, -5, 1, 161, 31, -8021, -14335, 686881, 2925631, -91860229, -583959551, 15741408385, 169511794271, -3832934048789, -54596554106879, 1106568438159809, 23024933751472255, -412744343093399429, -11208399032299519999, 177909311974519181281

Offset: 0

Seiichi Manyama, Oct 12 2017

sign

Robert Israel, Table of n, a(n) for n = 0..446

Cf. A111884, A293571, A293573.

f:= gfun:-rectoproc(n*(n+5)*(n+4)*(n+3)*(n+2)*(n+1)*a(n) + 2*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*a(n+1) + (n+5)*(n+4)*(n+3)*(8+3*n)*a(n+2) + (n+5)*(n+4)*(13+4*n)*a(n+3) + 3*(n+4)*(n+5)*a(n+4) + (9+2*n)*a(n+5) + a(n+6),
a(0) = 1, a(1) = 1, a(2) = -1, a(3) = -5, a(4) = 1, a(5) = 161}, a(n), remember):
map(f, [$0..25]); # Robert Israel, May 05 2020

N=66; x='x+O('x^N); Vec(serlaplace(exp(x/(1+x+x^2+x^3))))

N=66; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, exp(x^(4*k-3)-x^(4*k-2)))))

E.g.f.: Product_{k>0} exp(x^(4*k-3)) / exp(x^(4*k-2)).

n*(n+5)*(n+4)*(n+3)*(n+2)*(n+1)*a(n) + 2*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*a(n+1) + (n+5)*(n+4)*(n+3)*(8+3*n)*a(n+2) + (n+5)*(n+4)*(13+4*n)*a(n+3) + 3*(n+4)*(n+5)*a(n+4) + (9+2*n)*a(n+5) + a(n+6) = 0. - Robert Israel, May 05 2020

A317279 a(n) = Sum_{k=0..n} (-1)^(n-k)binomial(n-1,k-1)n^k*n!/k!.

Original entry on oeis.org

1, 1, 0, -9, -32, 225, 3456, 2695, -433152, -4495743, 47872000, 1768142871, 6703534080, -597265448351, -11959736205312, 126058380654375, 9454322092343296, 84694164336894465, -5776865438988238848, -192541299662555831753, 1511905067561779200000, 243338391925401706938081, 3972949090873574466519040

Offset: 0

Ilya Gutkovskiy, Jul 25 2018

sign

a(n) is the n-th term of the inverse Lah transform of the powers of n.

G. C. Greubel, Table of n, a(n) for n = 0..400
N. J. A. Sloane, Transforms
Index entries for sequences related to Laguerre polynomials

Cf. A111884, A293145, A317277, A317278.

l:= func< n, a, b | Evaluate(LaguerrePolynomial(n, a), b) >;
[1]cat[(-1)^(n+1)*Factorial(n)*l(n-1,1,n): n in [1..30]]; // G. C. Greubel, Mar 09 2021

A317279:= n -> `if`(n=0,1,(-1)^(n+1)*n!*simplify(LaguerreL(n-1,1,n), 'LaguerreL'));
seq(A317279(n), n = 0..30); # G. C. Greubel, Mar 09 2021

Join[{1}, Table[Sum[(-1)^(n-k) Binomial[n-1, k-1] n^k n!/k!, {k, n}], {n, 22}]]
Table[n! SeriesCoefficient[Exp[n x/(1 + x)], {x, 0, n}], {n, 0, 22}]
Table[n! SeriesCoefficient[Product[Exp[-n (-x)^k], {k, n}], {x, 0, n}], {n, 0, 22}]
Join[{1}, Table[(-1)^(n+1) n n! Hypergeometric1F1[1-n, 2, n], {n, 22}]]

a(n) = if (n==0, 1, (-1)^(n+1)*n!*pollaguerre(n-1, 1, n)); \\ Michel Marcus, Mar 10 2021

[1]+[(-1)^(n+1)*factorial(n)*gen_laguerre(n-1,1,n) for n in (1..30)] # G. C. Greubel, Mar 09 2021

a(n) = n! * [x^n] exp(n*x/(1 + x)).
a(n) = n! * [x^n] Product_{k>=1} exp(-n*(-x)^k).
a(n) = (-1)^(n+1) * n * n! * Hypergeometric1F1([1-n], [2], n) with a(0) = 1.
a(n) = (-1)^(n+1) * n! * LaguerreL(n-1, 1, n) with a(0) = 1. - G. C. Greubel, Mar 09 2021

A293120 Expansion of e.g.f. exp(x^2/(1+x)).

Original entry on oeis.org

1, 0, 2, -6, 36, -240, 1920, -17640, 183120, -2116800, 26943840, -374220000, 5628934080, -91122071040, 1579034096640, -29155689763200, 571308920582400, -11838533804697600, 258608278645516800, -5938673374272038400, 143003892952893772800

Offset: 0

Seiichi Manyama, Sep 30 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..444

Column k=1 of A293133.
Cf. A111884, A293121.
Cf. A052845.

CoefficientList[Series[E^(x^2/(1+x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2017 *)

x='x+O('x^66); Vec(serlaplace(exp(x^2/(1+x))))

E.g.f.: exp(x^2/(1+x)).
a(n) = (-1)^n * A052845(n).
a(n) ~ (-1)^n * n^(n-1/4) * exp(-3/2 + 2*sqrt(n) - n)/sqrt(2). - Vaclav Kotesovec, Sep 30 2017

cp's OEIS Frontend

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

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A383992 Series expansion of the exponential generating function exp(arbustive(x)) - 1 where arbustive(x) = (log(1+x) - x^2) / (1+x).

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A383993 Series expansion of the exponential generating function exp(tridup^!(x)) - 1 where tridup^!(x) = x / ((1+x) * (1+2*x)).

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A383994 Series expansion of the exponential generating function exp(wnp^!(x)) - 1 where wnp^!(x) = log(1+x) - x^2/(1+x).

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A383990 Series expansion of the exponential generating function exp(-dend(-x))-1 where dend(x) = (1 - sqrt(1+4*x)) / (2*x) + 1 (given by A000108).

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A293133 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^(k+1)/(1+x)).

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A293571 E.g.f.: exp(x/(1 + x + x^2)).

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Maple

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PARI

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A293572 E.g.f.: exp(x/(1 + x + x^2 + x^3)).

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Maple

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PARI

A383990 Series expansion of the exponential generating function exp(-dend(-x))-1 where dend(x) = (1 - sqrt(1+4x)) / (2x) + 1 (given by A000108).

A317279 a(n) = Sum_{k=0..n} (-1)^(n-k)binomial(n-1,k-1)n^k*n!/k!.