A351737 -id:A351737 - cp's OEIS frontend

A052506 Expansion of e.g.f. exp(x*exp(x)-x).

1, 0, 2, 3, 16, 65, 336, 1897, 11824, 80145, 586000, 4588001, 38239224, 337611001, 3144297352, 30779387745, 315689119456, 3383159052833, 37790736663456, 439036039824193, 5294386116882280, 66155074120062921, 855156188538926296, 11418964004032623809

Offset: 0

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

a(n) is the number of forests of rooted labeled trees with height exactly one. Equivalently, the number of idempotent mappings from {1,2,...,n} into {1,2,...,n} where each fixed point has at least one (other than itself) element mapped to it. See the second summation formula provided by Vladeta Jovovic with conditions on k, the number of fixed points. - Geoffrey Critzer, Sep 20 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 39
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Cf. A000248, A240989.

Cf. A351736, A351737.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x*Exp(x)-x) )); [Factorial(n-1)*b[n]: n in [1..m-1]]; // G. C. Greubel, May 13 2019

spec := [S,{S=Set(Tree), Tree=Prod(Z,Set(Z,0 < card))},labeled]: seq(combstruct[count](spec, size=n), n=0..20);

nn=20;Range[0,nn]! CoefficientList[Series[Exp[x(Exp[x]-1)], {x,0,nn}], x]  (* Geoffrey Critzer, Sep 20 2012 *)

my(x='x+O('x^30)); Vec(serlaplace( exp(x*exp(x)-x) )) \\ G. C. Greubel, Nov 15 2017

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(k-1)*x)^(k+1))) \\ Seiichi Manyama, Aug 29 2022

m = 30; T = taylor(exp(x*exp(x)-x), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 13 2019

a(n) = Sum_{k=0..n} binomial(n, k)*(n-k-1)^k. - Vladeta Jovovic, Apr 12 2003
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*k!*Stirling2(n-k, k). - Vladeta Jovovic, Dec 19 2004
a(n) ~ exp((1-r*(n+r))/(1+r)) * n^(n+1/2) * sqrt(1+r) / (r^n * sqrt((1+r)^3 + n*(1+3*r+r^2))), where r satisfies exp(r)*(1+r) - (1+n)/r = 1. - Vaclav Kotesovec, Aug 04 2014
(a(n)/n!)^(1/n) ~ exp(1/(2*LambertW(sqrt(n)/2))) / (2*LambertW(sqrt(n)/2)). - Vaclav Kotesovec, Aug 06 2014
G.f.: Sum_{k>=0} x^k / (1 - (k-1)*x)^(k+1). - Seiichi Manyama, Aug 29 2022

A351736 Expansion of e.g.f. exp( x * (exp(2 * x) - 1) ).

Original entry on oeis.org

1, 0, 4, 12, 80, 560, 4512, 40768, 407808, 4453632, 52605440, 667234304, 9032423424, 129822564352, 1972450443264, 31559866736640, 530043925495808, 9317136303718400, 170976603113127936, 3268020569256755200, 64928967058257346560, 1338431135849666052096

Offset: 0

Seiichi Manyama, May 20 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..488

Cf. A052506, A351737.

Cf. A053491, A351733.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(exp(2*x)-1))))

a(n) = n!*sum(k=0, n\2, 2^(n-k)*stirling(n-k, k, 2)/(n-k)!);

a(n) = sum(k=0, n, (2*k-1)^(n-k)*binomial(n, k)); \\ Seiichi Manyama, Aug 29 2022

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(2*k-1)*x)^(k+1))) \\ Seiichi Manyama, Aug 29 2022

a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-k) * Stirling2(n-k,k)/(n-k)!.
From Seiichi Manyama, Aug 29 2022: (Start)
a(n) = Sum_{k=0..n} (2*k-1)^(n-k) * binomial(n,k).
G.f.: Sum_{k>=0} x^k / (1 - (2*k-1)*x)^(k+1). (End)

A356806 a(n) = Sum_{k=0..n} (kn-1)^(n-k) binomial(n,k).

Original entry on oeis.org

1, 0, 4, 27, 448, 10625, 344736, 14437213, 753991680, 47974773393, 3650824000000, 326917384798301, 33956137832546304, 4041303651931462969, 545552768347831566336, 82828479894303251953125, 14040577418634835164921856, 2640293357854435329683551265

Offset: 0

Seiichi Manyama, Aug 29 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..274

Cf. A052506, A351736, A351737.

Cf. A356811, A356814, A356817.

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

a(n) = n!*sum(k=0, n\2, n^(n-k)*stirling(n-k, k, 2)/(n-k)!);

a(n) = n! * [x^n] exp( x * (exp(n * x) - 1) ).
a(n) = n! * Sum_{k=0..floor(n/2)} n^(n-k) * Stirling2(n-k,k)/(n-k)!.
a(n) = [x^n] Sum_{k>=0} x^k / (1 - (n*k-1)*x)^(k+1).

A356813 Expansion of e.g.f. exp(x * (1 - exp(3*x))).

Original entry on oeis.org

1, 0, -6, -27, 0, 1215, 12312, 45927, -657072, -15857937, -167699160, -266960529, 29356170984, 700068823623, 8419188469104, -1491045413265, -2856006296224992, -79065447339366945, -1162293393139510824, -744123842820101745, 538503788896323210360

Offset: 0

Seiichi Manyama, Aug 29 2022

sign

Seiichi Manyama, Table of n, a(n) for n = 0..470

Cf. A292893, A356812.

Cf. A351737, A356816.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-exp(3*x)))))

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (-x)^k/(1-(3*k+1)*x)^(k+1)))

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

a(n) = n!*sum(k=0, n\2, (-1)^k*3^(n-k)*stirling(n-k, k, 2)/(n-k)!);

G.f.: Sum_{k>=0} (-x)^k / (1 - (3*k+1)*x)^(k+1).
a(n) = Sum_{k=0..n} (-1)^k * (3*k+1)^(n-k) * binomial(n,k).
a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * 3^(n-k) * Stirling2(n-k,k)/(n-k)!.

A356816 Expansion of e.g.f. exp(-x * (exp(3*x) + 1)).

Original entry on oeis.org

1, -2, -2, 1, 88, 583, 676, -35597, -519392, -3359393, 19013884, 896435395, 13640180896, 85591357135, -1527872118356, -61100053650053, -1076294742932288, -7610985095240513, 200631806070276988, 9284475508083767059, 200226297062313730816, 1940767272243466116463

Offset: 0

Seiichi Manyama, Aug 29 2022

sign

Seiichi Manyama, Table of n, a(n) for n = 0..470

Cf. A356815, A356818.

Cf. A351737, A356813.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*(exp(3*x)+1))))

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (-x)^k/(1-(3*k-1)*x)^(k+1)))

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

G.f.: Sum_{k>=0} (-x)^k / (1 - (3*k-1)*x)^(k+1).

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

A351734 Expansion of e.g.f. exp( 3 * x * (exp(x) - 1) ).

Original entry on oeis.org

1, 0, 6, 9, 120, 555, 5148, 39711, 378528, 3715011, 39838260, 452684463, 5463506304, 69553644771, 930940368036, 13054086036855, 191222363275968, 2918620069099395, 46309955947643124, 762335523354333855, 12995722456718984160, 229045407317491457763

Offset: 0

Seiichi Manyama, May 20 2022

nonn

Cf. A052506, A351733.

Cf. A053490, A351737.

With[{nn=30},CoefficientList[Series[Exp[3x (Exp[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 02 2025 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(3*x*(exp(x)-1))))

a(n) = n!*sum(k=0, n\2, 3^k*stirling(n-k, k, 2)/(n-k)!);

a(n) = n! * Sum_{k=0..floor(n/2)} 3^k * Stirling2(n-k,k)/(n-k)!.

A354312 Expansion of e.g.f. exp( x/3 * (exp(3 * x) - 1) ).

Original entry on oeis.org

1, 0, 2, 9, 48, 315, 2496, 22491, 223728, 2437371, 28931040, 371291283, 5111412120, 75014135235, 1168157451384, 19228202401635, 333378840718944, 6069073767712587, 115683487658404272, 2303091818149762899, 47784447190060311240, 1031179733234906055507

Offset: 0

Seiichi Manyama, May 23 2022

sign

Cf. A052506, A354311.

Cf. A351734, A351737, A354310, A354314.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x/3*(exp(3*x)-1))))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*3^(j-2)*binomial(i-1, j-1)*v[i-j+1])); v;

a(n) = n!*sum(k=0, n\2, 3^(n-2*k)*stirling(n-k, k, 2)/(n-k)!);

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

a(n) = n! * Sum_{k=0..floor(n/2)} 3^(n-2*k) * Stirling2(n-k,k)/(n-k)!.

A356827 Expansion of e.g.f. exp(x * exp(3*x)).

Original entry on oeis.org

1, 1, 7, 46, 361, 3436, 37729, 463366, 6280369, 93015352, 1491337441, 25684077706, 472217487625, 9221588527204, 190441412508481, 4143470377262806, 94663498086222049, 2264440394856702832, 56570146384760433217, 1472545685988162638722

Offset: 0

Seiichi Manyama, Aug 29 2022

nonn

Cf. A000248, A003725, A216689, A295552.

Cf. A277456, A336951, A351737, A355501, A356820.

A356827 := proc(n)
    add((3*k)^(n-k) * binomial(n,k),k=0..n) ;
end proc:
seq(A356827(n),n=0..70) ; # R. J. Mathar, Dec 04 2023

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x*exp(3*x))))

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-3*k*x)^(k+1)))

a(n) = sum(k=0, n, (3*k)^(n-k)*binomial(n, k));

G.f.: Sum_{k>=0} x^k / (1 - 3*k*x)^(k+1).

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

A367886 Expansion of e.g.f. 1/(1 - x * (exp(3*x) - 1)).

Original entry on oeis.org

1, 0, 6, 27, 324, 3645, 54918, 923643, 18061704, 394663833, 9607469130, 256997250279, 7502660832780, 237243300445125, 8079508278302958, 294800526215739315, 11473728720705019152, 474469344621574172721, 20774758472643152149650

Offset: 0

Seiichi Manyama, Dec 04 2023

nonn

Cf. A052848, A367885.

Cf. A337039, A351737.

a(n) = n!*sum(k=0, n\2, 3^(n-k)*k!*stirling(n-k, k, 2)/(n-k)!);

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

a(n) = n! * Sum_{k=0..floor(n/2)} 3^(n-k) * k! * Stirling2(n-k,k)/(n-k)!.

A362839 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} k^(n-j) * Stirling2(n-j,j)/(n-j)!.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 3, 0, 1, 0, 6, 12, 16, 0, 1, 0, 8, 27, 80, 65, 0, 1, 0, 10, 48, 216, 560, 336, 0, 1, 0, 12, 75, 448, 2025, 4512, 1897, 0, 1, 0, 14, 108, 800, 5120, 21708, 40768, 11824, 0, 1, 0, 16, 147, 1296, 10625, 67584, 260253, 407808, 80145, 0

Offset: 0

Seiichi Manyama, May 05 2023

			Square array begins:
  1,  1,   1,    1,    1,     1, ...
  0,  0,   0,    0,    0,     0, ...
  0,  2,   4,    6,    8,    10, ...
  0,  3,  12,   27,   48,    75, ...
  0, 16,  80,  216,  448,   800, ...
  0, 65, 560, 2025, 5120, 10625, ...

Columns k=0..3 give: A000007, A052506, A351736, A351737.
Main diagonal gives A356806.
Cf. A362652.

T(n, k) = n!*sum(j=0, n\2, k^(n-j)*stirling(n-j, j, 2)/(n-j)!);

E.g.f. of column k: exp(x * (exp(k * x) - 1)).
G.f. of column k: Sum_{j>=0} x^j / (1 - (k*j-1)*x)^(j+1).
T(n,k) = Sum_{j=0..n} (k*j-1)^(n-j) * binomial(n,j).

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A052506 Expansion of e.g.f. exp(x*exp(x)-x).

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A351736 Expansion of e.g.f. exp( x * (exp(2 * x) - 1) ).

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A356806 a(n) = Sum_{k=0..n} (k*n-1)^(n-k) * binomial(n,k).

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A356813 Expansion of e.g.f. exp(x * (1 - exp(3*x))).

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A356816 Expansion of e.g.f. exp(-x * (exp(3*x) + 1)).

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A351734 Expansion of e.g.f. exp( 3 * x * (exp(x) - 1) ).

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A354312 Expansion of e.g.f. exp( x/3 * (exp(3 * x) - 1) ).

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A356806 a(n) = Sum_{k=0..n} (kn-1)^(n-k) binomial(n,k).