A052873 -id:A052873 - cp's OEIS frontend

A052857 A simple grammar. a(n)=n*A052873(n-1).

0, 1, 2, 15, 184, 3145, 68976, 1846999, 58413440, 2130740721, 88061420800, 4066862460991, 207556068584448, 11600364266112505, 704664527894104064, 46226086991634882375, 3256882066245640093696, 245279323467051422886241

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

G. C. Greubel, Table of n, a(n) for n = 0..250
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 825
Index entries for sequences related to Laguerre polynomials

[n lt 2 select n else n*Factorial(n-2)*Evaluate(LaguerrePolynomial(n-2, 1), -n): n in [0..20]]; // G. C. Greubel, Feb 23 2021

spec := [S,{C=Set(B),S=Prod(Z,C),B=Sequence(S,1<= card)},labeled]:
seq(combstruct[count](spec,size=n), n=0..20);
# Alternatively:
a := n -> `if`(n<2,n, n!*hypergeom([-n+2],[2],-n));
seq(simplify(a(n)), n=0..17); # Peter Luschny, Apr 20 2016

Table[If[0<=n<=1, n, (n-1)! Sum[(n^k Binomial[n-2, k-1])/k!, {k,n-1}]], {n,0,20}] (* Michael De Vlieger, Apr 20 2016 *)
Table[If[n<2, n, n*(n-2)!*LaguerreL[n-2, 1, -n]], {n, 0, 20}] (* G. C. Greubel, Feb 23 2021 *)

a(n):=if n=1 then 1 else ((n-1)!*sum((n^k*binomial(n-2,k-1))/k!,k,1,n-1)); /* Vladimir Kruchinin, May 10 2011 */

[n if n<2 else n*factorial(n-2)*gen_laguerre(n-2, 1, -n) for n in (0..20)] # G. C. Greubel, Feb 23 2021

E.g.f.: exp(RootOf(exp(_Z)*x*_Z+exp(_Z)*x-_Z))*x.
a(n) = (n-1)!*Sum_{k=1..n-1} n^k*binomial(n-2,k-1)/k!, a(1)=1. - Vladimir Kruchinin, May 10 2011
a(n) = n!*hypergeom([-n+2], [2], -n) for n>=2. - Peter Luschny, Apr 20 2016
a(n) ~ exp(n/phi - n) * phi^(2*n-2) * n^(n-1) / 5^(1/4), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 01 2017
E.g.f. A(x) satisfies: A(x) = x*exp(A(x)/(1 - A(x))). - Ilya Gutkovskiy, Apr 04 2019
a(n) = n * (n-2)! * LaguerreL(n-2, 1, -n) with a(0) = 0 and a(1) = 1. - G. C. Greubel, Feb 23 2021

A088690 E.g.f.: A(x) = f(xA(x)), where f(x) = (1+x)exp(x).

Original entry on oeis.org

1, 2, 11, 106, 1489, 27696, 643579, 17973488, 586899009, 21953140480, 925890264331, 43480125312768, 2250352192663249, 127280062346049536, 7811329076598534075, 517016126622623635456, 36713034605774835974401, 2784127167066690618458112

Offset: 0

Paul D. Hanna, Oct 06 2003

nonn

Radius of convergence of A(x): r = tau^2*exp(-tau) = 0.20588... and A(r) = (1+tau)*exp(tau), where tau=(sqrt(5)-1)/2 and r = limit a(n)/a(n+1)*n as n->infinity.

Alois P. Heinz, Table of n, a(n) for n = 0..200

a := n -> n!*simplify(hypergeom([-n], [2], -n-1)):
seq(a(n), n=0..15); # Peter Luschny, Apr 20 2016

CoefficientList[1/x*InverseSeries[Series[x*E^(-x)/(1+x), {x, 0, 21}], x],x]*Range[0, 20]! (* Vaclav Kotesovec, Jan 24 2014 *)

a(n)=n!*polcoeff(((1+x)*exp(x))^(n+1)+x*O(x^n),n,x)/(n+1)

a(n) = n! * [x^n] ((1+x)*exp(x))^(n+1)/(n+1).
a(n) = Sum_{k=1..n} n^(k-2)*n!/k!*binomial(n-1,k-1) (offset 1). - Vladeta Jovovic, Jun 17 2006
E.g.f.: A(x) = (1/x)*series_reversion(x*exp(-x)/(1+x)). - Paul D. Hanna, Jun 17 2006
E.g.f.: B(x)/(1-x*B(x)), where B(x) is e.g.f. for A052873(). - Vladeta Jovovic, Jun 18 2006
a(n) ~ 5^(-1/4) * ((1+sqrt(5))/2)^(2*n+2) * exp((sqrt(5) - 1 - (3 - sqrt(5))*n)/2) * n^(n-1). - Vaclav Kotesovec, Jan 24 2014
a(n) = n!*hypergeom([-n], [2], -n-1). - Peter Luschny, Apr 20 2016

A361093 E.g.f. satisfies A(x) = exp( 1/(1 - x * A(x)^2) - 1 ).

Original entry on oeis.org

1, 1, 7, 97, 2049, 58541, 2114143, 92419965, 4746108769, 280105517881, 18683156508471, 1389960074426969, 114119472522112225, 10249863809271551973, 999746622121255094479, 105236583967331849218741, 11891012005206169120252737, 1435560112909007680593616625

Offset: 0

Seiichi Manyama, Mar 01 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..334

Cf. A052873, A361094, A361095, A361096, A361097.

Cf. A212722, A361065.

Table[n! * Sum[(2*n+1)^(k-1) * Binomial[n-1,n-k]/k!, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 02 2023 *)

a(n) = n!*sum(k=0, n, (2*n+1)^(k-1)*binomial(n-1, n-k)/k!);

a(n) = n! * Sum_{k=0..n} (2*n+1)^(k-1) * binomial(n-1,n-k)/k!.

a(n) ~ n^(n-1) / (2 * 3^(1/4) * (2 - sqrt(3))^n * exp((2 - sqrt(3))*n - (sqrt(3) - 1)/2)). - Vaclav Kotesovec, Mar 02 2023

A380663 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x) * exp(-x/(1 - x)) ).

Original entry on oeis.org

1, 2, 15, 208, 4285, 117936, 4075099, 169736960, 8282604537, 463604723200, 29287449579751, 2061571190059008, 160023548976361525, 13580237335641417728, 1250935473495646861875, 124307671411309327876096, 13255531892787507819759601, 1509841440567809574906101760

Offset: 0

Seiichi Manyama, Jan 30 2025

nonn

Index entries for reversions of series

Cf. A377831, A380664.

Cf. A052873, A380665.

a(n) = n!*sum(k=0, n, (n+1)^(k-1)*binomial(2*n, n-k)/k!);

E.g.f. A(x) satisfies A(x) = exp(x * A(x)/(1 - x*A(x)))/(1 - x*A(x)).

a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(2*n,n-k)/k!.

A361094 E.g.f. satisfies A(x) = exp( 1/(1 - x * A(x)^3) - 1 ).

Original entry on oeis.org

1, 1, 9, 166, 4717, 182136, 8911549, 528571408, 36864033945, 2956595372416, 268116203622961, 27128338649300736, 3029974270053623941, 370289278173654092800, 49150116757136815109733, 7041536364582774222616576, 1083004122024520209576760369

Offset: 0

Seiichi Manyama, Mar 01 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..322

Cf. A052873, A361093, A361095, A361096, A361097.

Cf. A212917, A361066.

Table[n! * Sum[(3*n+1)^(k-1) * Binomial[n-1,n-k]/k!, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 02 2023 *)

a(n) = n!*sum(k=0, n, (3*n+1)^(k-1)*binomial(n-1, n-k)/k!);

a(n) = n! * Sum_{k=0..n} (3*n+1)^(k-1) * binomial(n-1,n-k)/k!.

a(n) ~ (5 + sqrt(21))^n * n^(n-1) / (3^(3/4) * 7^(1/4) * 2^n * exp((3 - sqrt(21))/6 + (5 - sqrt(21))*n/2)). - Vaclav Kotesovec, Mar 02 2023

A380665 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^2 * exp(-x/(1 - x)) ).

Original entry on oeis.org

1, 3, 31, 586, 16401, 612336, 28678231, 1618268688, 106946168769, 8105456425600, 693228400344591, 66055574392722432, 6940237183385667409, 797165049089377683456, 99381018789002592800775, 13365207839280075801020416, 1928719845703457066672384769, 297293268794967068206087176192

Offset: 0

Seiichi Manyama, Jan 30 2025

nonn

Index entries for reversions of series

Cf. A052873, A380663.

Cf. A377832.

a(n) = n!*sum(k=0, n, (n+1)^(k-1)*binomial(3*n+1, n-k)/k!);

E.g.f. A(x) satisfies A(x) = exp(x * A(x)/(1 - x*A(x)))/(1 - x*A(x))^2.

a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(3*n+1,n-k)/k!.

A361095 E.g.f. satisfies A(x) = exp( 1/(1 - x/A(x)) - 1 ).

Original entry on oeis.org

1, 1, 1, -2, -3, 56, -155, -2736, 34489, 72064, -6599799, 53676800, 1155350581, -32238425088, -3604716947, 14790925735936, -235482791871375, -4972572910452736, 254158358486634001, -1028499606209101824, -202204782754527137939, 5371925138905661440000

Offset: 0

Seiichi Manyama, Mar 01 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..415

Cf. A052873, A361093, A361094, A361096, A361097.

a(n) = if(n==0, 1, n!*sum(k=1, n, (-n+1)^(k-1)*binomial(n-1, n-k)/k!));

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

A361096 E.g.f. satisfies A(x) = exp( 1/(1 - x/A(x)^2) - 1 ).

Original entry on oeis.org

1, 1, -1, 1, 17, -339, 4999, -63587, 566145, 3549241, -405637489, 15518099961, -446235202799, 9617693853925, -75522664207017, -7341781870733099, 596513949276803969, -30104875035438797583, 1144712508931072057375, -27381639204739332379151

Offset: 0

Seiichi Manyama, Mar 01 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..385

Cf. A052873, A361093, A361094, A361095, A361097.

a(n) = n!*sum(k=0, n, (-2*n+1)^(k-1)*binomial(n-1, n-k)/k!);

a(n) = n! * Sum_{k=0..n} (-2*n+1)^(k-1) * binomial(n-1,n-k)/k!.

A361097 E.g.f. satisfies A(x) = exp( 1/(1 - x/A(x)^3) - 1 ).

Original entry on oeis.org

1, 1, -3, 22, -251, 3816, -71207, 1542640, -36997431, 929097856, -22062115979, 334968255744, 13395424571725, -2177817789105152, 201597999475333329, -16622491076645341184, 1332634806870147259537, -107073894723559010304000

Offset: 0

Seiichi Manyama, Mar 01 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..359

Cf. A052873, A361093, A361094, A361095, A361096.

a(n) = n!*sum(k=0, n, (-3*n+1)^(k-1)*binomial(n-1, n-k)/k!);

a(n) = n! * Sum_{k=0..n} (-3*n+1)^(k-1) * binomial(n-1,n-k)/k!.

A361142 E.g.f. satisfies A(x) = exp( xA(x)^2/(1 - xA(x)) ).

Original entry on oeis.org

1, 1, 7, 91, 1773, 46401, 1529593, 60911103, 2845757449, 152663425633, 9250206248781, 624880915165959, 46569571425664477, 3795729136868379777, 335902071304953561073, 32074779600414913885231, 3287242849289861637185937, 359917016243351870997841473

Offset: 0

Seiichi Manyama, Mar 02 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..338

Cf. A000262, A361143.

Cf. A052868, A052873, A161630.

Table[n! * Sum[(n+k+1)^(k-1) * Binomial[n-1,n-k]/k!, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 03 2023 *)

a(n) = n!*sum(k=0, n, (n+k+1)^(k-1)*binomial(n-1, n-k)/k!);

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

a(n) ~ s^2 * sqrt((2 - r*s)/(2 + r*s*(-2 + s*(2 - r*s)^2))) * n^(n-1) / (exp(n) * r^(n - 1/2)), where r = 0.14220768719194290600038416000340972911571484385125... and s = 1.549730657609106944767484487465870359529391502493... are roots of the system of equations exp(r*s^2/(1 - r*s)) = s, r*s^2*(2 - r*s) = (1 - r*s)^2. - Vaclav Kotesovec, Mar 03 2023

cp's OEIS Frontend

A052857 A simple grammar. a(n)=n*A052873(n-1).

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A088690 E.g.f.: A(x) = f(x*A(x)), where f(x) = (1+x)*exp(x).

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A361093 E.g.f. satisfies A(x) = exp( 1/(1 - x * A(x)^2) - 1 ).

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A380663 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x) * exp(-x/(1 - x)) ).

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A361094 E.g.f. satisfies A(x) = exp( 1/(1 - x * A(x)^3) - 1 ).

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Mathematica

PARI

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A380665 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^2 * exp(-x/(1 - x)) ).

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PARI

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A361095 E.g.f. satisfies A(x) = exp( 1/(1 - x/A(x)) - 1 ).

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A361096 E.g.f. satisfies A(x) = exp( 1/(1 - x/A(x)^2) - 1 ).

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A088690 E.g.f.: A(x) = f(xA(x)), where f(x) = (1+x)exp(x).

A361142 E.g.f. satisfies A(x) = exp( xA(x)^2/(1 - xA(x)) ).