A003725 -id:A003725 - cp's OEIS frontend

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

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]

A215652 Exponential Riordan array [exp(x*exp(-x)),x].

Original entry on oeis.org

1, 1, 1, -1, 2, 1, -2, -3, 3, 1, 9, -8, -6, 4, 1, -4, 45, -20, -10, 5, 1, -95, -24, 135, -40, -15, 6, 1, 414, -665, -84, 315, -70, -21, 7, 1, 49, 3312, -2660, -224, 630, -112, -28, 8, 1, -10088, 441, 14904, -7980, -504, 1134, -168, -36, 9, 1

Offset: 0

Peter Bala, Sep 11 2012

For commuting lower unitriangular matrices A and B we define A raised to the matrix power B, denoted A^^B, to be the matrix Exp(B*Log(A)). Here Exp denotes the matrix exponential and the matrix logarithm Log(A) is defined as sum {n >= 1} (-1)^(n+1)*(A-1)^n/n. Call the present triangle X and let P denote Pascal's triangle A007318. Then X solves the matrix equation X^^P = P. Equivalently, the infinite tower of matrix powers X^^(X^^(X^^(....))) equals P. Note that the infinite tower of powers P^^(P^^(P^^(...))) of the Pascal triangle equals the hyperbinomial array A088956. Thus we might view the present array as the hypobinomial triangle.

			Triangle begins
.n\k.|....0.....1.....2.....3.....4.....5.....6.....7
= = = = = = = = = = = = = = = = = = = = = = = = = = =
..0..|....1
..1..|....1.....1
..2..|...-1.....2.....1
..3..|...-2....-3.....3.....1
..4..|....9....-8....-6.....4.....1
..5..|...-4....45...-20...-10.....5.....1
..6..|..-95...-24...135...-40...-15.....6.....1
..7..|..414..-665...-84...315...-70...-21.....7.....1
...

G. Helms, Pascalmatrix tetrated

Cf. A003506, A003725 (column 0), A007318, A088956.

max = 9; MapIndexed[ Take[#1, #2[[1]]]&, CoefficientList[ Series[ Exp[x*t]*Exp[x*Exp[-x]], {x, 0, max}, {t, 0, max}], {x, t}]*Range[0, max]!, 1] // Flatten (* Jean-François Alcover, Jan 08 2013 *)

T(n,k) = binomial(n,k)*A003725(n-k).
The triangle equals P^^Q, where P is Pascal's triangle and Q is the inverse of P. Column 0 equals A003725.
E.g.f.: exp(x*t)*exp(x*exp(-x)) = 1 + (1 + t)*x + (-1 + 2*t + t^2)*x^2/2! + (-2 - 3*t + 3*t^2 + t^3)*x^3/3! + ....
The infinitesimal generator for this triangle is the generalized exponential Riordan array [x*exp(-x),x], which factors as [x,x]*[exp(-x),x] = A132440*A007318^(-1). The infinitesimal generator begins
..0
..1....0
.-2....2....0
..3...-6....3....0
.-4...12..-12....4....0
This is a signed version of the triangle of denominators from Leibniz's harmonic triangle - see A003506.

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).

A345652 Expansion of the e.g.f. exp(-1 + (x + 1)*exp(-x)).

Original entry on oeis.org

1, 0, -1, 2, 0, -16, 65, -78, -749, 6232, -22068, -28920, 1004685, -7408740, 22263215, 157632230, -2874256740, 21590948480, -53087332675, -956539294506, 16344490525835, -132605481091060, 294656170409328, 9113173803517344, -167298122286332823

Offset: 0

Mélika Tebni, Jun 21 2021

sign

For all p prime, a(p)/(p-1) == 1 (mod p). - Mélika Tebni, Mar 21 2022

			exp(-1+(x+1)*exp(-x)) = 1 - x^2/2! + 2*x^3/3! - 16*x^5/5! + 65*x^6/6! - 78*x^7/7! - 749*x^8/8! + 6232*x^9/9! + ...

Seiichi Manyama, Table of n, a(n) for n = 0..560

Cf. A003725, A014182, A327006.
Cf. A292935 (without 1+x: EGF e^(e^(-x)-1)), A000110 (absolute values: Bell numbers, EGF e^(e^x-1))
Cf. A003725, A106828, A347210, A347571.

a := series(exp(-1+(x+1)*exp(-x)), x=0, 25): seq(n!*coeff(a, x, n), n=0..24);
a := proc(n) option remember; `if`(n=0, 1, add((n-1)*binomial(n-2, k)*(-1)^(n-1-k)*a(k), k=0..n-2)) end: seq(a(n), n=0..24);
# third program:
A345652 := n -> add((-1)^(n-k)*combinat[bell](k)*A106828(n, k), k=0..iquo(n, 2)):
seq(A345652(n), n=0..24); # Mélika Tebni, Sep 21 2021

nmax = 24; CoefficientList[Series[Exp[-1+(x+1)*Exp[-x]], {x, 0, nmax}], x] Range[0, nmax]!

seq(n) = {Vec(serlaplace(exp(-1+(x+1)*exp(-x + O(x*x^n)))))} \\ Andrew Howroyd, Jun 21 2021

a(n) = if(n==0, 1, sum(k=2, n, (-1)^(k-1)*(k-1)*binomial(n-1, k-1)*a(n-k))); \\ Seiichi Manyama, Mar 15 2022

The e.g.f. y(x) satisfies y' = -x*y*exp(-x).
a(n) = Sum_{k=0..n-2} (n-1)*binomial(n-2, k)*(-1)^(n-1-k)*a(k) for n > 0.
Conjecture: a(n) = 0 for only n = 1 and n = 4.
Conjecture: For all p prime, a(p)^2 == 1 (mod p).
Stronger conjecture: For n > 1, a(n) == -1 (mod n) iff n is a prime or 6. - M. F. Hasler, Jun 23 2021
a(n) = Sum_{k=0..floor(n/2)} (-1)^(n-k)*Bell(k)*A106828(n, k). - Mélika Tebni, Sep 21 2021
a(n) = Sum_{k=0..n} (-1)^k*A003725(n-k)*Bell(k)*binomial(n, k). - Mélika Tebni, Mar 21 2022

A318365 Expansion of e.g.f. exp(x*exp(-x)/(1 - x)).

Original entry on oeis.org

1, 1, 1, 4, 21, 116, 805, 6504, 59353, 608320, 6901641, 85824080, 1160786341, 16959401304, 266133942061, 4463567862376, 79669223849265, 1507610621184224, 30145968665822737, 635066714078714016, 14057275047440540221, 326159212986987669640, 7915118313077599105461, 200503241124736099689656

Offset: 0

Ilya Gutkovskiy, Aug 24 2018

nonn

Cf. A000166, A000240, A000262, A003725, A318364.

seq(n!*coeff(series(exp(x*exp(-x)/(1-x)),x=0,24),x,n),n=0..23); # Paolo P. Lava, Jan 09 2019

nmax = 23; CoefficientList[Series[Exp[x Exp[-x]/(1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[k Subfactorial[k - 1] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]

x = 'x + O('x^25); Vec(serlaplace(exp(x*exp(-x)/(1 - x)))) \\ Michel Marcus, Aug 25 2018

a(0) = 1; a(n) = Sum_{k=1..n} A000240(k)*binomial(n-1,k-1)*a(n-k).

a(n) ~ exp(exp(-1)/2 - 1/4 + 2*exp(-1/2)*sqrt(n) - n) * n^(n - 1/4) / sqrt(2). - Vaclav Kotesovec, Aug 25 2018

A346748 E.g.f.: exp( (x * exp(-x) + sinh(x)) / 2 ).

Original entry on oeis.org

1, 1, 0, 0, 4, -1, -9, 103, -132, -535, 7731, -25117, -18072, 1078215, -6917039, 16312667, 186611792, -2454241183, 14370311311, 1436259867, -934228834216, 10658996229479, -54990712418263, -185381404760729, 7270919988375200, -80130195880201583, 391992372213719679

Offset: 0

Ilya Gutkovskiy, Aug 01 2021

sign

Exponential transform of A001057.

Cf. A001057, A003724, A003725, A346746, A346747.

nmax = 26; CoefficientList[Series[Exp[(x Exp[-x] + Sinh[x])/2], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) Binomial[n - 1, k - 1] Floor[(k + 1)/2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 26}]

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A001057(k) * a(n-k).

Typo in a(26) corrected by Georg Fischer, Nov 30 2021

A347432 E.g.f.: exp( exp(x) * (exp(x) - 1 - x) ).

Original entry on oeis.org

1, 0, 1, 4, 14, 66, 397, 2626, 18797, 148238, 1281134, 11943790, 118998365, 1262189748, 14203022537, 168835162632, 2111832477426, 27708387132906, 380355066174121, 5449577398256414, 81316095965242989, 1261149374033472626, 20293627142875917978, 338263983223664609198

Offset: 0

Ilya Gutkovskiy, Sep 02 2021

nonn

Exponential transform of A000295.

Cf. A000295, A000296, A003725, A055882, A143405, A347434, A347435.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*(2^j-j-1), j=1..n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 02 2021

nmax = 23; CoefficientList[Series[Exp[Exp[x] (Exp[x] - 1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] (2^k - k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A000295(k) * a(n-k).
a(n) = Sum_{k=0..n} (-1)^k * binomial(n,k) * A003725(k) * A143405(n-k).
a(n) ~ n^(n + 1/2) * (exp(exp(r)*(exp(r) - r - 1) - r/2 - n) / (r^(n + 1/2) * sqrt(2*exp(r)*(1 + 2*r) - (2 + r*(4 + r))))), where r = LambertW(n)/2 + (4 + LambertW(n)) * LambertW(n)^(3/2) / (8 * sqrt(n) * (1 + LambertW(n))). - Vaclav Kotesovec, Jul 07 2022

A358063 Expansion of e.g.f. exp( x * exp(-x^3) ).

Original entry on oeis.org

1, 1, 1, 1, -23, -119, -359, 1681, 38641, 269137, 599761, -22461119, -347288039, -2477852519, 13993475497, 670329026641, 8887630708321, 29011883003041, -1682765787379679, -40673626173010943, -409560067877703479, 4061870252008891561, 235100528524188216121

Offset: 0

Seiichi Manyama, Oct 29 2022

Cf. A003725, A357948.

Cf. A354553.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*exp(-x^3))))

a(n) = n!*sum(k=0, n\3, (-n+3*k)^k/(k!*(n-3*k)!));

a(n) = n! * Sum_{k=0..floor(n/3)} (-n + 3*k)^k/(k! * (n - 3*k)!).

A320258 a(n) = n! * [x^n] exp(xexp(-nx)).

Original entry on oeis.org

1, 1, -3, 10, 81, -4724, 156205, -4406814, 76958273, 3775676248, -698309272899, 72802616429830, -6310377003297455, 435451735391849892, -10028808876450831571, -4757293711381352201774, 1464955115044140633346305, -310063138309576689774123728, 55179706013436631385620675837

Offset: 0

Ilya Gutkovskiy, Oct 08 2018

sign

Cf. A003725, A295552.

Table[n! SeriesCoefficient[Exp[x Exp[-n x]], {x, 0, n}], {n, 0, 18}]
Table[SeriesCoefficient[Sum[x^k/(1 + n k x)^(k + 1), {k, 0, n}], {x, 0, n}], {n, 0, 18}]
Join[{1}, Table[Sum[Binomial[n, k] (-n k)^(n - k), {k, 0, n}], {n, 18}]]

a(n) = [x^n] Sum_{k>=0} x^k/(1 + n*k*x)^(k+1).

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A215652 Exponential Riordan array [exp(x*exp(-x)),x].

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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).

Views

Author

Keywords

Comments

Links

Crossrefs

A345652 Expansion of the e.g.f. exp(-1 + (x + 1)*exp(-x)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A318365 Expansion of e.g.f. exp(x*exp(-x)/(1 - x)).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A346748 E.g.f.: exp( (x * exp(-x) + sinh(x)) / 2 ).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

Extensions

A347432 E.g.f.: exp( exp(x) * (exp(x) - 1 - x) ).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

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

A320258 a(n) = n! * [x^n] exp(xexp(-nx)).