A194689 -id:A194689 - cp's OEIS frontend

A217924 a(n) = n! * [x^n] exp(2*exp(x) - x - 2). Row sums of triangle A217537.

1, 1, 3, 9, 35, 153, 755, 4105, 24323, 155513, 1064851, 7760745, 59895203, 487397849, 4166564147, 37298443977, 348667014723, 3395240969785, 34365336725715, 360837080222761, 3923531021460707, 44108832866004121, 511948390801374835, 6126363766802713481

Offset: 0

Peter Luschny, Oct 15 2012

nonn

The inverse binomial transform of a(n) is A194689.
A087981(n) = Sum_{k=0..n} (-1)^k*s(n+1,k+1)*a(k);
|A000023(n)| = |Sum_{k=0..n} (-1)^(n-k)*s(n,k)*a(k)|
where s(n,k) are the unsigned Stirling numbers of first kind.
a(n) is the number of inequivalent set partitions of {1,2,...,n} where two blocks are considered equivalent when one can be obtained from the other by an alternating (even) permutation. - Geoffrey Critzer, Mar 17 2013

			a(3)=9 because we have: {1,2,3}; {1,3,2}; {1}{2,3}; {1}{3,2}; {2}{1,3}; {2}{3,1}; {3}{1,2}; {3}{2,1}; {1}{2}{3}. - _Geoffrey Critzer_, Mar 17 2013

Vaclav Kotesovec, Table of n, a(n) for n = 0..556

Similar recurrences: A124758, A243499, A284005, A329369, A341392, A372205.

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(2*Exp(x) -x-2) ))); // G. C. Greubel, Jan 09 2025

egf := exp(2*exp(x) - x - 2): ser := series(egf, x, 25):
seq(n!*coeff(ser, x, n), n = 0..23);  # Peter Luschny, Apr 22 2024

nn=23;Range[0,nn]!CoefficientList[Series[Exp[2 Exp[x]-x-2],{x,0,nn}],x]  (* Geoffrey Critzer, Mar 17 2013 *)
nmax = 25; CoefficientList[Series[1/(1 - x + ContinuedFractionK[-2*k*x^2 , 1 - (k + 1)*x, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 25 2017 *)

a(n):=sum(sum(binomial(n,k-j)*2^j*(-1)^(k-j)*stirling2(n-k+j,j),j,0,k),k,0,n); /* Vladimir Kruchinin, Feb 28 2015 */

def A217924_list(n):
    T = A217537_triangle(n)
    return [add(T.row(n)) for n in range(n)]
A217924_list(24)

def A217924_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(2*exp(x)-x-2) ).egf_to_ogf().list()
print(A217924_list(40)) # G. C. Greubel, Jan 09 2025

G.f.: 1/Q(0) where Q(k) = 1 + x*k - x/(1 - 2*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 06 2013
E.g.f.: exp(2*exp(x) - x - 2). -  Geoffrey Critzer, Mar 17 2013
G.f.: 1/Q(0), where Q(k) = 1 - (k+1)*x - 2*(k+1)*x^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 03 2013
G.f.: T(0)/(1-x), where T(k) = 1 - 2*x^2*(k+1)/( 2*x^2*(k+1) - (1-x-x*k)*(1-2*x-x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 19 2013
a(n) = Sum_{k=0..n} Sum_{j=0..k} binomial(n,k-j)*2^j*(-1)^(k-j)*Stirling2(n-k+j,j). - Vladimir Kruchinin, Feb 28 2015
a(n) = exp(-2) * Sum_{k>=0} 2^k * (k - 1)^n / k!. - Ilya Gutkovskiy, Jun 27 2020
Conjecture: a(n) = Sum_{k=0..2^n-1} A372205(k). - Mikhail Kurkov, Nov 21 2021 [Rewritten by Peter Luschny, Apr 22 2024]
a(n) ~ 2 * n^(n-1) * exp(n/LambertW(n/2) - n - 2) / (sqrt(1 + LambertW(n/2)) * LambertW(n/2)^(n-1)). - Vaclav Kotesovec, Jun 26 2022

Name extended by a formula of Geoffrey Critzer by Peter Luschny, Apr 22 2024

A339014 E.g.f.: exp(2 * (exp(x) - 1 - x - x^2 / 2)).

Original entry on oeis.org

1, 0, 0, 2, 2, 2, 42, 142, 366, 3082, 18626, 86990, 596158, 4485626, 30214498, 224897662, 1871664190, 15587540042, 134045407458, 1231183979886, 11725017784574, 114812031304986, 1170100796863202, 12371771640238174, 134796972965052350, 1514854948728869354

Offset: 0

Ilya Gutkovskiy, Nov 19 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..563

Cf. A001861, A006505, A194689.

nmax = 25; CoefficientList[Series[Exp[2 (Exp[x] - 1 - x - x^2/2)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 3, n}]; Table[a[n], {n, 0, 25}]

my(x='x+O('x^30)); Vec(serlaplace(exp(2 * (exp(x) - 1 - x - x^2/2)))) \\ Michel Marcus, Nov 19 2020

a(0) = 1; a(n) = 2 * Sum_{k=3..n} binomial(n-1,k-1) * a(n-k).

a(n) = Sum_{k=0..n} binomial(n,k) * A006505(k) * A006505(n-k).

A339017 E.g.f.: exp(2 * (exp(x) - 1 - x - x^2 / 2 - x^3 / 6)).

Original entry on oeis.org

1, 0, 0, 0, 2, 2, 2, 2, 142, 506, 1346, 3170, 53198, 375234, 1880738, 7919082, 72104190, 678488362, 5164781154, 33220643026, 271431061614, 2710340281426, 26278673924322, 228727591600826, 2081516848032222, 21560234032116154, 236863265302626722, 2521687569105476002

Offset: 0

Ilya Gutkovskiy, Nov 19 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..569

Cf. A001861, A057837, A194689, A339014, A339027.

nmax = 27; CoefficientList[Series[Exp[2 (Exp[x] - 1 - x - x^2/2 - x^3/6)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 4, n}]; Table[a[n], {n, 0, 27}]

my(x='x+O('x^30)); Vec(serlaplace(exp(2*(exp(x) - 1 - x - x^2/2 - x^3/6)))) \\ Michel Marcus, Nov 19 2020

a(0) = 1; a(n) = 2 * Sum_{k=4..n} binomial(n-1,k-1) * a(n-k).

a(n) = Sum_{k=0..n} binomial(n,k) * A057837(k) * A057837(n-k).

A339027 E.g.f.: exp(2 * (exp(x) - 1 - x - x^2 / 2 - x^3 / 6 - x^4 / 24)).

Original entry on oeis.org

1, 0, 0, 0, 0, 2, 2, 2, 2, 2, 506, 1850, 5018, 12014, 26886, 1066782, 8193070, 42723722, 185108514, 719359762, 10426744914, 118490840686, 976376930502, 6583701431086, 38977418758494, 377188932759354, 4671829781287922, 51479602726372402, 483303800325691922

Offset: 0

Ilya Gutkovskiy, Nov 20 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..576

Cf. A001861, A057814, A194689, A339014, A339017.

nmax = 28; CoefficientList[Series[Exp[2 (Exp[x] - 1 - x - x^2/2 - x^3/6 - x^4/24)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 5, n}]; Table[a[n], {n, 0, 28}]

my(x='x+O('x^30)); Vec(serlaplace(exp(2 * (exp(x) - 1 - x - x^2/2 - x^3/6 - x^4/24)))) \\ Michel Marcus, Nov 20 2020

a(0) = 1; a(n) = 2 * Sum_{k=5..n} binomial(n-1,k-1) * a(n-k).

a(n) = Sum_{k=0..n} binomial(n,k) * A057814(k) * A057814(n-k).

A355247 Expansion of e.g.f. exp(2*(exp(x) - 1 + x)).

Original entry on oeis.org

1, 4, 18, 90, 494, 2946, 18926, 130066, 950654, 7353794, 59954638, 513333618, 4601380766, 43062556322, 419742815726, 4252083713874, 44680229906622, 486145710591874, 5468499473222670, 63503107472489266, 760281866742088670, 9373065303624742498, 118858898763010225198

Offset: 0

Vaclav Kotesovec, Jun 25 2022

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..550

Cf. A000110, A001861, A035009, A194689, A217924, A293024, A339014.

nmax = 25; CoefficientList[Series[Exp[2*Exp[x]-2+2*x], {x, 0, nmax}], x] * Range[0, nmax]!
Table[(BellB[n+2, 2] - BellB[n+1, 2])/4, {n, 0, 25}] (* Vaclav Kotesovec, Jul 21 2025 *)

a(n) ~ n^(n+2) * exp(n/LambertW(n/2) - n - 2) / (4 * sqrt(1 + LambertW(n/2)) * LambertW(n/2)^(n+2)).

a(n) = Sum_{k=0..n} binomial(n,k) * Bell(k+1) * Bell(n-k+1). - Ilya Gutkovskiy, Jun 26 2022

A355253 Expansion of e.g.f. exp(2(exp(x) - 1) - 3x).

Original entry on oeis.org

1, -1, 3, -5, 19, -29, 171, -69, 2339, 5139, 57563, 303403, 2397011, 17237507, 139011211, 1151110299, 10076637827, 91903924979, 874688607035, 8656097294091, 88932728790195, 946748093175523, 10426787247224043, 118620906668843131, 1392128306377939427, 16833088095308098003

Offset: 0

Vaclav Kotesovec, Jun 26 2022

sign

Inverse binomial transform of A194689.

Vaclav Kotesovec, Table of n, a(n) for n = 0..555

Cf. A001861, A217923, A194689, A355252.

nmax = 30; CoefficientList[Series[Exp[2*Exp[x]-2-3*x], {x, 0, nmax}], x] * Range[0, nmax]!

my(x='x+O('x^30)); Vec(serlaplace(exp(2*(exp(x) - 1) - 3*x))) \\ Michel Marcus, Dec 04 2023

a(n) ~ 8 * n^(n-3) * exp(n/LambertW(n/2) - n - 2) / (sqrt(1 + LambertW(n/2)) * LambertW(n/2)^(n-3)).

a(0) = 1; a(n) = -3 * a(n-1) + 2 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - Ilya Gutkovskiy, Dec 04 2023

A367888 Expansion of e.g.f. exp(3(exp(x) - 1) - 2x).

Original entry on oeis.org

1, 1, 4, 13, 61, 304, 1747, 10945, 74830, 550687, 4335109, 36272086, 320980645, 2991373597, 29253607780, 299258487553, 3193634980753, 35469069928792, 409082335024591, 4890313138089133, 60489400453642822, 772967507343358171, 10189818916331129017, 138398721137005215526

Offset: 0

Ilya Gutkovskiy, Dec 04 2023

nonn

Cf. A000296, A027710, A126617, A194689, A355254, A367889.

b:= proc(n, k, m) option remember; `if`(n=0, 3^m, `if`(k>0,
      b(n-1, k-1, m+1)*k, 0)+m*b(n-1, k, m)+b(n-1, k+1, m))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Apr 29 2025

nmax = 23; CoefficientList[Series[Exp[3 (Exp[x] - 1) - 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -2 a[n - 1] + 3 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]
Table[Sum[Binomial[n, k] (-2)^(n - k) BellB[k, 3], {k, 0, n}], {n, 0, 23}]

my(x='x+O('x^30)); Vec(serlaplace(exp(3*(exp(x) - 1) - 2*x))) \\ Michel Marcus, Dec 04 2023

G.f. A(x) satisfies: A(x) = 1 - x * ( 2 * A(x) - 3 * A(x/(1 - x)) / (1 - x) ).
a(n) = exp(-3) * Sum_{k>=0} 3^k * (k-2)^n / k!.
a(0) = 1; a(n) = -2 * a(n-1) + 3 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * (-2)^(n-k) * A027710(k).

A367891 Expansion of e.g.f. exp(4*(exp(x) - 1 - x)).

Original entry on oeis.org

1, 0, 4, 4, 52, 164, 1364, 7620, 60148, 449252, 3831700, 33811716, 320082228, 3178774564, 33234163668, 363535920196, 4153091085172, 49406896240996, 610777358429204, 7830140410294148, 103914148870277556, 1425254885630973604, 20173671034640405588

Offset: 0

Ilya Gutkovskiy, Dec 04 2023

nonn

Cf. A000296, A078944, A194689, A346739, A367890.

nmax = 22; CoefficientList[Series[Exp[4 (Exp[x] - 1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 4 Sum[Binomial[n - 1, k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 22}]
Table[Sum[Binomial[n, k] (-4)^(n - k) BellB[k, 4], {k, 0, n}], {n, 0, 22}]

my(x='x+O('x^30)); Vec(serlaplace(exp(4*(exp(x) - 1 - x)))) \\ Michel Marcus, Dec 04 2023

G.f. A(x) satisfies: A(x) = 1 - 4 * x * ( A(x) - A(x/(1 - x)) / (1 - x) ).
a(n) = exp(-4) * Sum_{k>=0} 4^k * (k-4)^n / k!.
a(0) = 1; a(n) = 4 * Sum_{k=1..n-1} binomial(n-1,k) * a(n-k-1).
a(n) = Sum_{k=0..n} binomial(n,k) * (-4)^(n-k) * A078944(k).

A367890 Expansion of e.g.f. exp(3*(exp(x) - 1 - x)).

Original entry on oeis.org

1, 0, 3, 3, 30, 93, 633, 3342, 22809, 156063, 1183872, 9453711, 80455125, 721576560, 6809391111, 67332650007, 695777512638, 7493572404345, 83926492573341, 975467527353750, 11744536832206149, 146234590864310019, 1880198749437144456, 24928860500681953683

Offset: 0

Ilya Gutkovskiy, Dec 04 2023

nonn

Cf. A000296, A027710, A194689, A346738, A355253, A355254, A367888, A367891.

nmax = 23; CoefficientList[Series[Exp[3 (Exp[x] - 1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 3 Sum[Binomial[n - 1, k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]
Table[Sum[Binomial[n, k] (-3)^(n - k) BellB[k, 3], {k, 0, n}], {n, 0, 23}]

my(x='x+O('x^30)); Vec(serlaplace(exp(3*(exp(x) - 1 - x)))) \\ Michel Marcus, Dec 04 2023

G.f. A(x) satisfies: A(x) = 1 - 3 * x * ( A(x) - A(x/(1 - x)) / (1 - x) ).
a(n) = exp(-3) * Sum_{k>=0} 3^k * (k-3)^n / k!.
a(0) = 1; a(n) = 3 * Sum_{k=1..n-1} binomial(n-1,k) * a(n-k-1).
a(n) = Sum_{k=0..n} binomial(n,k) * (-3)^(n-k) * A027710(k).

A367920 Expansion of e.g.f. exp(4(exp(x) - 1) - 2x).

Original entry on oeis.org

1, 2, 8, 36, 196, 1196, 8116, 60108, 481140, 4126540, 37671540, 364068172, 3707910772, 39645022540, 443540780660, 5177560304972, 62903920321140, 793654042136908, 10378403752717940, 140413475790402892, 1962339063781284468, 28287778534523140428, 420059992540347885172

Offset: 0

Ilya Gutkovskiy, Dec 04 2023

nonn

Cf. A000296, A078944, A126617, A194689, A367888, A367891, A367919, A367921.

nmax = 22; CoefficientList[Series[Exp[4 (Exp[x] - 1) - 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -2 a[n - 1] + 4 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

G.f. A(x) satisfies: A(x) = 1 - 2 * x * ( A(x) - 2 * A(x/(1 - x)) / (1 - x) ).
a(n) = exp(-4) * Sum_{k>=0} 4^k * (k-2)^n / k!.
a(0) = 1; a(n) = -2 * a(n-1) + 4 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).

cp's OEIS Frontend

A217924 a(n) = n! * [x^n] exp(2*exp(x) - x - 2). Row sums of triangle A217537.

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A339014 E.g.f.: exp(2 * (exp(x) - 1 - x - x^2 / 2)).

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A339017 E.g.f.: exp(2 * (exp(x) - 1 - x - x^2 / 2 - x^3 / 6)).

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A339027 E.g.f.: exp(2 * (exp(x) - 1 - x - x^2 / 2 - x^3 / 6 - x^4 / 24)).

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A355247 Expansion of e.g.f. exp(2*(exp(x) - 1 + x)).

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A355253 Expansion of e.g.f. exp(2*(exp(x) - 1) - 3*x).

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A367888 Expansion of e.g.f. exp(3*(exp(x) - 1) - 2*x).

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A367891 Expansion of e.g.f. exp(4*(exp(x) - 1 - x)).

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A355253 Expansion of e.g.f. exp(2(exp(x) - 1) - 3x).

A367888 Expansion of e.g.f. exp(3(exp(x) - 1) - 2x).

A367920 Expansion of e.g.f. exp(4(exp(x) - 1) - 2x).