A086331 -id:A086331 - cp's OEIS frontend

A362349 a(n) = n! * Sum_{k=0..floor(n/4)} k^k / (k! * (n-4*k)!).

1, 1, 1, 1, 25, 121, 361, 841, 82321, 728785, 3633841, 13313521, 2195435881, 28125394441, 196393341145, 981274727161, 227100486456481, 3807339471993121, 34186011461595361, 216366574074187105, 64438384450412161081, 1335035336388170601241

Offset: 0

Seiichi Manyama, Apr 17 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..431
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A086331, A362347, A362348.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)/(1+lambertw(-x^4))))

E.g.f.: exp(x) / (1 + LambertW(-x^4)).

A204042 The number of functions f:{1,2,...,n}->{1,2,...,n} (endofunctions) such that all of the fixed points in f are isolated.

Original entry on oeis.org

1, 1, 2, 12, 120, 1520, 23160, 413952, 8505280, 197631072, 5125527360, 146787894440, 4601174623584, 156693888150384, 5761055539858528, 227438694372072120, 9596077520725211520, 430920897407809702208, 20520683482765477749120, 1032920864149903149579336, 54797532208320308334631840

Offset: 0

Geoffrey Critzer, Jan 09 2012

nonn

Note this sequence counts the functions enumerated by A065440 for which the statement is vacuously true.

a(n) is also the number of partial endofunctions on {1,2,...,n} without fixed points.

			a(2)=2 because there are two functions f:{1,2}->{1,2} in which all the fixed points are isolated: 1->1,2->2  and 1->2,2->1 (which has no fixed points).

Alois P. Heinz, Table of n, a(n) for n = 0..386

Cf. A065440, A086331, A350134.

Row sums of A349454.

a:= n-> add((j-1)^j*binomial(n, j), j=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Dec 16 2021

t = Sum[n^(n-1) x^n/n!, {n,1,20}]; Range[0,20]! CoefficientList[Series[Exp[x] Exp[Log[1/(1-t)]-t], {x,0,20}], x]

E.g.f.: exp(x)*A(x) where A(x) is the e.g.f. for A065440.
a(n) ~ exp(exp(-1)-1)*n^n. - Vaclav Kotesovec, Sep 24 2013
a(n) = Sum_{j=0..n} (j-1)^j * binomial(n,j). - Alois P. Heinz, Dec 16 2021

A277462 E.g.f.: cos(x)/(1 + LambertW(-x)).

Original entry on oeis.org

1, 1, 3, 24, 233, 2860, 42875, 758856, 15488657, 358164432, 9254769459, 264273873600, 8264362186489, 280896392748608, 10310601442639147, 406479520869636480, 17129450693008029729, 768404013933189112064, 36557893891263190204259, 1838650651518153170939904

Offset: 0

Vaclav Kotesovec, Oct 16 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A000312, A086331, A277461, A277464.

CoefficientList[Series[Cos[x]/(1+LambertW[-x]), {x, 0, 25}], x] * Range[0, 25]!
Table[Cos[Pi*n/2] + Sum[Binomial[n, k] * Cos[Pi*(n-k)/2] * k^k, {k, 1, n}], {n, 0, 25}] (* Vaclav Kotesovec, Oct 28 2016 *)

x='x+O('x^50); Vec(serlaplace(cos(x)/(1 + lambertw(-x)))) \\ G. C. Greubel, Nov 07 2017

a(n) ~ cos(exp(-1)) * n^n.

A277467 E.g.f.: tan(x)/(1+LambertW(-x)).

Original entry on oeis.org

0, 1, 2, 14, 116, 1376, 19926, 346128, 6964712, 159396352, 4085415850, 115906440704, 3605365584732, 121998144397312, 4461190462108030, 175305587376883712, 7366747721719011280, 329646098258032459776, 15649117182518598570834, 785528920149992297070592

Offset: 0

Vaclav Kotesovec, Oct 16 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A000312, A086331, A277468.

CoefficientList[Series[Tan[x]/(1+LambertW[-x]), {x, 0, 25}], x] * Range[0, 25]!
Table[Sin[Pi*n/2] * 2^(n+1) * (2^(n+1) - 1) * BernoulliB[n+1] / (n+1) + Sum[Binomial[n, k] * Sin[Pi*k/2] * 2^(k+1) * (2^(k+1)-1) * BernoulliB[k+1] /(k+1) * (n-k)^(n-k), {k, 0, n-1}], {n, 0, 25}] (* Vaclav Kotesovec, Oct 28 2016 *)

x='x+O('x^30); concat([0], Vec(serlaplace(tan(x)/(1 + lambertw(-x))))) \\ G. C. Greubel, May 29 2018

a(n) ~ tan(exp(-1)) * n^n.

A277469 E.g.f.: arcsin(x)/(1 + LambertW(-x)).

Original entry on oeis.org

0, 1, 2, 13, 112, 1329, 19344, 336533, 6778752, 155247777, 3980956800, 112984562813, 3515475849216, 118984054897681, 4351800687259648, 171034439586509445, 7188243684485414912, 321696219477456836929, 15273278824827215388672, 766732102664665113137517

Offset: 0

Vaclav Kotesovec, Oct 16 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A000312, A086331, A277470.

CoefficientList[Series[ArcSin[x]/(1+LambertW[-x]), {x, 0, 25}], x] * Range[0, 25]!
Flatten[{0, Table[(1-(-1)^n)/2 * (n-2)!!^2 + Sum[Binomial[n, k] * (1-(-1)^k)/2 * (k-2)!!^2 * (n-k)^(n-k), {k, 1, n-1}], {n, 1, 25}]}] (* Vaclav Kotesovec, Oct 28 2016 *)

x='x+O('x^50); concat([0], Vec(serlaplace(asin(x)/(1 + lambertw(-x))))) \\ G. C. Greubel, Nov 07 2017

a(n) ~ arcsin(exp(-1)) * n^n.

A277474 E.g.f.: -exp(-x)*LambertW(-x).

Original entry on oeis.org

0, 1, 0, 6, 36, 380, 4830, 74382, 1342712, 27825912, 651274650, 16994464850, 489240628932, 15404364096420, 526634857318934, 19428038813967630, 769280055136105200, 32543192449030871792, 1464827827285673677746, 69903432558329996409642, 3525344776953738276010940

Offset: 0

Vaclav Kotesovec, Oct 17 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A000169, A069856, A086331, A277473.

CoefficientList[Series[-Exp[-x]*LambertW[-x], {x, 0, 20}], x] * Range[0, 20]!
Table[Sum[(-1)^(n-k)*Binomial[n, k]*k^(k-1), {k, 1, n}], {n, 0, 20}]

x='x+O('x^50); concat([0], Vec(serlaplace(-exp(-x)*lambertw(-x)))) \\ G. C. Greubel, Nov 07 2017

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

a(n) ~ exp(-exp(-1)) * n^(n-1).

A343899 a(n) = Sum_{k=0..n} (k!)^k * binomial(n,k).

Original entry on oeis.org

1, 2, 7, 232, 332669, 24884861086, 139314218808181027, 82606412229102532926819812, 6984964247802365417561163907914436537, 109110688415634181158572146813823590758078301022074, 395940866122426284350759726810156652343313286283891529199276099071

Offset: 0

Seiichi Manyama, May 03 2021

nonn

Binomial transform of (n!)^n.

Seiichi Manyama, Table of n, a(n) for n = 0..30

Cf. A000522, A036740, A046662, A086331, A343898, A343900, A343928.

a[n_] := Sum[(k!)^k * Binomial[n, k], {k, 0, n}]; Array[a, 11, 0] (* Amiram Eldar, May 05 2021 *)

a(n) = sum(k=0, n, k!^k*binomial(n, k));

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k!*x)^k/(1-x)^(k+1)))

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=0, N, k!^(k-1)*x^k)))

G.f.: Sum_{k>=0} (k! * x)^k/(1 - x)^(k+1).

E.g.f.: exp(x) * Sum_{k>=0} (k!)^(k-1) * x^k.

A277457 E.g.f.: exp(2*x)/(1+LambertW(-x)).

Original entry on oeis.org

1, 3, 12, 71, 616, 7197, 105052, 1829291, 36922928, 846851993, 21744781684, 617832652527, 19242299657896, 651815827343189, 23857403245171724, 938247816632341043, 39455261828928309088, 1766645684585351990961, 83913998998426051745764, 4214295288128637488870327, 223120214856875472660345176

Offset: 0

Vaclav Kotesovec, Oct 16 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A086331, A277454, A277456, A277485.

CoefficientList[Series[Exp[2*x]/(1+LambertW[-x]), {x, 0, 20}], x]*Range[0, 20]!
Table[1 + Sum[Binomial[n, m]*(1 + Sum[Binomial[m, k]*k^k, {k, 1, m}]), {m, 1, n}], {n, 0, 20}]
Table[2^n + Sum[Binomial[n, k]*2^(n-k)*k^k, {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 28 2016 *)

x='x+O('x^50); Vec(serlaplace(exp(2*x)/(1 + lambertw(-x)))) \\ G. C. Greubel, Nov 07 2017

a(n) ~ exp(2*exp(-1)) * n^n.

A277465 Expansion of e.g.f. log(1+x)/(1 + LambertW(-x)).

Original entry on oeis.org

0, 1, 1, 11, 86, 1084, 15654, 275113, 5548024, 127423728, 3272008650, 92988690893, 2896148079516, 98104636748468, 3590611928294286, 141201205469361945, 5937400341113630032, 265833516437952849024, 12625912572901413474834, 634047172218326393377149

Offset: 0

Vaclav Kotesovec, Oct 16 2016

nonn

Robert Israel, Table of n, a(n) for n = 0..386

Cf. A000312, A086331, A277466.

S:= series(log(1+x)/(1+LambertW(-x)), x, 51):
seq(coeff(S,x,n)*n!, n=0..50); # Robert Israel, Oct 26 2016

CoefficientList[Series[Log[1+x]/(1+LambertW[-x]), {x, 0, 25}], x] * Range[0, 25]!

x='x+O('x^50); concat([0],Vec(serlaplace(log(1+x)/(1 + lambertw(-x))))) \\ G. C. Greubel, Nov 07 2017

E.g.f.: log(1+x)/(1 + LambertW(-x)).
a(n) ~ log(1+exp(-1)) * n^n.
a(n) = (-1)^(n+1)*(n-1)! + Sum_{j=1..n-1} a(j)*binomial(n,j)*(n-j)^(n-j-1). - Robert Israel, Oct 26 2016

A277466 E.g.f.: -log(1-x)/(1+LambertW(-x)).

Original entry on oeis.org

0, 1, 3, 17, 146, 1684, 24294, 419383, 8412760, 192078864, 4914973770, 139265564723, 4327699948956, 146323675764044, 5347193667136398, 210005149832116455, 8820722263274822992, 394546588041904397184, 18723398414958791004690, 939550079246853331267203

Offset: 0

Vaclav Kotesovec, Oct 16 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A000312, A086331, A277465.

CoefficientList[Series[-Log[1-x]/(1+LambertW[-x]), {x, 0, 25}], x] * Range[0, 25]!
Flatten[{0, Table[(n-1)! + n!*Sum[k^k/(k!*(n-k)), {k, 1, n-1}], {n, 1, 25}]}] (* Vaclav Kotesovec, Oct 28 2016 *)

x='x+O('x^50); concat([0], Vec(serlaplace(-log(1-x)/(1 + lambertw(-x))))) \\ G. C. Greubel, Nov 07 2017

a(n) ~ -log(1-exp(-1)) * n^n.

cp's OEIS Frontend

A362349 a(n) = n! * Sum_{k=0..floor(n/4)} k^k / (k! * (n-4*k)!).

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A204042 The number of functions f:{1,2,...,n}->{1,2,...,n} (endofunctions) such that all of the fixed points in f are isolated.

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A277462 E.g.f.: cos(x)/(1 + LambertW(-x)).

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Mathematica

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A277467 E.g.f.: tan(x)/(1+LambertW(-x)).

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Mathematica

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A277469 E.g.f.: arcsin(x)/(1 + LambertW(-x)).

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Mathematica

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A277474 E.g.f.: -exp(-x)*LambertW(-x).

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A343899 a(n) = Sum_{k=0..n} (k!)^k * binomial(n,k).

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A277457 E.g.f.: exp(2*x)/(1+LambertW(-x)).

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