A006153 -id:A006153 - cp's OEIS frontend

A351761 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} k^(n-j) * (n-j)^j/j!.

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 12, 21, 0, 1, 4, 24, 102, 148, 0, 1, 5, 40, 279, 1160, 1305, 0, 1, 6, 60, 588, 4332, 16490, 13806, 0, 1, 7, 84, 1065, 11536, 84075, 281292, 170401, 0, 1, 8, 112, 1746, 25220, 282900, 1958058, 5598110, 2403640, 0

Offset: 0

Seiichi Manyama, Feb 18 2022

			Square array begins:
  1,    1,     1,     1,      1,      1, ...
  0,    1,     2,     3,      4,      5, ...
  0,    4,    12,    24,     40,     60, ...
  0,   21,   102,   279,    588,   1065, ...
  0,  148,  1160,  4332,  11536,  25220, ...
  0, 1305, 16490, 84075, 282900, 746525, ...

Columns k=0..3 give A000007, A006153, A351762, A351763.
Main diagonal gives A351765.
Cf. A292860, A351776.

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

T(n, k) = if(n==0, 1, k*n*sum(j=0, n-1, binomial(n-1, j)*T(j, k)));

E.g.f. of column k: 1/(1 - k*x*exp(x)).

T(0,k) = 1 and T(n,k) = k * n * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.

A358738 Expansion of Sum_{k>=0} k! * ( x/(1 - k*x) )^k.

Original entry on oeis.org

1, 1, 3, 15, 103, 893, 9341, 114355, 1603155, 25318137, 444689497, 8597568671, 181430298479, 4149361409077, 102229328244837, 2699254206069387, 76038064580742091, 2276259442660623857, 72160287650141753777, 2414950992007231422007

Offset: 0

Seiichi Manyama, Nov 29 2022

nonn

Cf. A001339, A006153, A080108.

Cf. A358740, A358741.

nmax = 20; CoefficientList[Series[Sum[k! * (x/(1 - k*x))^k, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 18 2023 *)

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

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

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

a(n) ~ n! / ((1 + LambertW(1))^2 * LambertW(1)^n). - Vaclav Kotesovec, Feb 18 2023

A332408 a(n) = Sum_{k=0..n} binomial(n,k) * k! * k^n.

Original entry on oeis.org

1, 1, 10, 213, 8284, 513105, 46406286, 5772636373, 945492503320, 197253667623681, 51069324556151290, 16067283861476491941, 6037615013420387657844, 2670812587802323522405393, 1373842484756310928089102022, 813119045938378747809030359445

Offset: 0

Ilya Gutkovskiy, Apr 23 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..232

Cf. A000522, A006153, A031971, A063170, A072034, A256016, A277452, A332627.

Join[{1}, Table[Sum[Binomial[n, k] k! k^n, {k, 0, n}], {n, 1, 15}]]

a(n) = sum(k=0, n, binomial(n,k) * k! * k^n); \\ Michel Marcus, Apr 24 2020

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*x*exp(x))^k))) \\ Seiichi Manyama, Feb 19 2022

G.f.: Sum_{k>=0} k! * k^k * x^k / (1 - k*x)^(k+1).
a(n) = n! * Sum_{k=0..n} k^n / (n-k)!.
a(n) ~ c * n! * n^n, where c = A073229 = exp(exp(-1)). - Vaclav Kotesovec, Feb 20 2021
E.g.f.: Sum_{k>=0} (k*x*exp(x))^k. - Seiichi Manyama, Feb 19 2022

A336952 E.g.f.: 1 / (1 - x * exp(4*x)).

Original entry on oeis.org

1, 1, 10, 102, 1336, 22200, 443664, 10334128, 275060608, 8236914048, 274069953280, 10031110907136, 400520747437056, 17324601073921024, 807023462798608384, 40278407730378332160, 2144307919689898491904, 121291661335680615284736, 7264376142168665821741056

Offset: 0

Ilya Gutkovskiy, Aug 08 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..375

Column k=4 of A351790.

Cf. A002697, A006153, A235328, A326324, A328183, A336950, A336951.

nmax = 18; CoefficientList[Series[1/(1 - x Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[n! Sum[(4 (n - k))^k/k!, {k, 0, n}], {n, 1, 18}]]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k 4^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

seq(n)={ Vec(serlaplace(1 / (1 - x*exp(4*x + O(x^n))))) } \\ Andrew Howroyd, Aug 08 2020

a(n) = n! * Sum_{k=0..n} (4 * (n-k))^k / k!.
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k * 4^(k-1) * a(n-k).
a(n) ~ n! * (4/LambertW(4))^n / (1 + LambertW(4)). - Vaclav Kotesovec, Aug 09 2021

A346893 Expansion of e.g.f. 1 / (1 - x^5 * exp(x) / 5!).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 6, 21, 56, 126, 504, 6006, 67320, 577863, 4038034, 24975951, 165481680, 1553590220, 19495772856, 249507077436, 2910465717648, 31103684847837, 326286335505438, 3766644374319673, 51399738264984648, 785038533451101930

Offset: 0

Ilya Gutkovskiy, Aug 06 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..498

Column k=5 of A351703.
Cf. A006153, A346888, A346889, A346890.
Cf. A000389, A145455.

nmax = 25; CoefficientList[Series[1/(1 - x^5 Exp[x]/5!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[k, 5] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 25}]

my(x='x+O('x^25)); Vec(serlaplace(1/(1-x^5*exp(x)/5!))) \\ Michel Marcus, Aug 06 2021

a(n) = n!*sum(k=0, n\5, k^(n-5*k)/(120^k*(n-5*k)!)); \\ Seiichi Manyama, May 13 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * binomial(k,5) * a(n-k).
a(n) ~ n! / ((1 + LambertW(2^(3/5)*3^(1/5)/5^(4/5))) * 5^(n+1) * LambertW(2^(3/5)*3^(1/5)/5^(4/5))^n). - Vaclav Kotesovec, Aug 08 2021
a(n) = n! * Sum_{k=0..floor(n/5)} k^(n-5*k)/(120^k * (n-5*k)!). - Seiichi Manyama, May 13 2022

A358081 Expansion of e.g.f. 1/(1 - x^3 * exp(x)).

Original entry on oeis.org

1, 0, 0, 6, 24, 60, 840, 10290, 80976, 847224, 13306320, 190271070, 2677088040, 46082426676, 874515884424, 16582066303530, 336875275380000, 7539189088358640, 176554878235711776, 4295134487197296054, 111114287924643309240, 3036073975138066955820

Offset: 0

Seiichi Manyama, Oct 30 2022

Seiichi Manyama, Table of n, a(n) for n = 0..431

Cf. A006153, A358080.

Cf. A292889, A355575, A358065.

With[{nn=30},CoefficientList[Series[1/(1-x^3 Exp[x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 12 2025 *)

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

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

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

a(n) ~ n! / ((1 + LambertW(1/3)) * 3^(n+1) * LambertW(1/3)^n). - Vaclav Kotesovec, Oct 30 2022

A364981 E.g.f. satisfies A(x) = 1 + xA(x)exp(x*A(x)^3).

Original entry on oeis.org

1, 1, 4, 39, 580, 11685, 298566, 9248701, 336886936, 14112113049, 668422303210, 35325208755441, 2060811941835780, 131547166492534117, 9120279070776381886, 682489450793082237285, 54828316394224735284016, 4706545644403274325580593

Offset: 0

Seiichi Manyama, Aug 15 2023

nonn

Cf. A006153, A161633, A364938, A364980.

Join[{1}, Table[n! * Sum[k^(n-k) * Binomial[3*n-2*k+1,k] / ((3*n-2*k+1)*(n-k)!), {k,0,n}], {n,1,20}]] (* Vaclav Kotesovec, Nov 18 2023 *)

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

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

a(n) ~ sqrt((1 + r*s^3)/(12*s + 9*r*s^4)) * n^(n-1) / (exp(n) * r^(n + 1/2)), where r = 0.1811100305436879929789759231994897963241226689807... and s = 1.522012903517407628213363540403002787906223513104... are real roots of the system of equations 1 + exp(r*s^3)*r*s = s, 3*r*s^3*(s-1) = 1. - Vaclav Kotesovec, Nov 18 2023

A380050 E.g.f. A(x) satisfies A(x) = sqrt( 1 + 2xexp(x)*A(x) ).

Original entry on oeis.org

1, 1, 3, 9, 25, 25, -429, -4151, -8175, 320625, 5241475, 23329801, -705579159, -18521117303, -150119840493, 3366485315145, 138253031778721, 1780881865542625, -28047359274759549, -1854674541474191351, -34985197604145203655, 332608115115937927161

Offset: 0

Seiichi Manyama, Jan 11 2025

sign

Cf. A006153, A380051.

Cf. A297010, A380044.

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

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

E.g.f.: exp( arcsinh(x*exp(x)) ).
E.g.f.: x*exp(x) + sqrt(1 + x^2*exp(2*x)).
a(n) = n! * Sum_{k=0..n} 2^k * k^(n-k) * binomial(k/2+1/2,k)/( (k+1)*(n-k)! ).

A380051 E.g.f. A(x) satisfies A(x) = ( 1 + 3xexp(x)*A(x) )^(1/3).

Original entry on oeis.org

1, 1, 2, 1, -12, -15, 526, 1617, -49608, -302111, 8126010, 85724001, -2020009628, -34232466255, 696686324166, 18267485751985, -310973114236944, -12533263924965183, 168118610439268594, 10727427541319225793, -100693940482485604260, -11178369799980253348079

Offset: 0

Seiichi Manyama, Jan 11 2025

sign

Cf. A006153, A380050.

Cf. A380047.

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

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

A199673 Number of ways to form k labeled groups, each with a distinct leader, using n people. Triangle T(n,k) = n!*k^(n-k)/(n-k)! for 1 <= k <= n.

Original entry on oeis.org

1, 2, 2, 3, 12, 6, 4, 48, 72, 24, 5, 160, 540, 480, 120, 6, 480, 3240, 5760, 3600, 720, 7, 1344, 17010, 53760, 63000, 30240, 5040, 8, 3584, 81648, 430080, 840000, 725760, 282240, 40320, 9, 9216, 367416, 3096576, 9450000, 13063680, 8890560, 2903040, 362880

Offset: 1

Dennis P. Walsh, Nov 08 2011

T(n,1)=n since there are n choices for the leader of the single group. Also, T(n,n)=n! since each of the n groups consist solely of a leader and there are n! ways to assign the n people to the n labeled groups.
In general, T(n,k) = n!*k^(n-k)/(n-k)! since there are n!/(n-k)! ways to assign leaders to the k labeled groups and there are k^(n-k) ways to map the remaining (n-k) people to the k groups.
T(n,k) is the number of functions of [n] to an arbitrary k-subset of [n], where each of the k target values is used at least once.
The number of ways to distribute n different toys among k girls and k boys to that each girl gets exactly one toy. - Dennis P. Walsh, Sep 10 2012

			T(3,2)=12 since there are 12 ways to form group 1 and group 2, both with leaders, using people p1, p2, and p3, as illustrated below. The leader will be denoted Lj if person pj is designated the leader of the group.
Group 1   Group 2
{L1,p2}   {L3}
{L1,p3}   {L2}
{L1}      {L2,p3}
{L1}      {p2,L3}
{L2,p1}   {L3}
{L2,p3}   {L1}
{L2}      {L1,p3}
{L2}      {p1,L3}
{L3,p2}   {L1}
{L3,p1}   {L2}
{L3}      {L1,p2}
{L3}      {p1,L2}
First rows of triangle T(n,k):
  1;
  2,    2;
  3,   12,      6;
  4,   48,     72,      24;
  5,  160,    540,     480,     120;
  6,  480,   3240,    5760,    3600,      720;
  7, 1344,  17010,   53760,   63000,    30240,    5040;
  8, 3584,  81648,  430080,  840000,   725760,  282240,   40320;
  9, 9216, 367416, 3096576, 9450000, 13063680, 8890560, 2903040, 362880;

Joerg Arndt, Table of n, a(n) for n = 1..561
George Kesidis, Takis Konstantopoulos, and Michael A. Zazanis, Age of information without service preemption, arXiv:2104.08050 [cs.PF], 2021.
Dennis P. Walsh, Assigning people into labeled groups with leaders
Dennis P. Walsh, Toy Story 2

[Factorial(n)*k^(n-k)/Factorial(n-k): k in [1..n], n in [1..9]];  // Bruno Berselli, Nov 09 2011

seq(seq(n!*k^(n-k)/(n-k)!, k=1..n), n=1..9);

nn = 10; a = y x Exp[x]; f[list_] := Select[list, # > 0 &]; Drop[Map[f, Range[0, nn]! CoefficientList[Series[1/(1 - a) , {x, 0, nn}], {x, y}]], 1] // Flatten  (* Geoffrey Critzer, Jan 21 2012 *)

T(n,k)=n!*k^(n-k)/(n-k)!;
/* print triangle: */
for (n=1, 15, for (k=1,n, print1(T(n,k),", ")); print() );
/* Joerg Arndt, Sep 21 2012 */

T(n,k) = n!*k^(n-k)/(n-k)! = k!*k^(n-k)*binomial(n,k) for 1 <= k <= n.
E.g.f.: (x*e^x)^k,for fixed k.
T(n,k1+k2) = Sum_{j=0..n} binomial(n,j)*T(j,k1)*T(n-j,k2).
T(n,1) = A000027(n);
T(n,2) = A001815(n);
T(n,3) = A052791(n);
Sum_{k=1..n} T(n,k) = A006153(n).
T(n,n) = A000142(n) = n!. - Dennis P. Walsh, Sep 10 2012

cp's OEIS Frontend

A351761 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} k^(n-j) * (n-j)^j/j!.

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A358738 Expansion of Sum_{k>=0} k! * ( x/(1 - k*x) )^k.

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A332408 a(n) = Sum_{k=0..n} binomial(n,k) * k! * k^n.

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A336952 E.g.f.: 1 / (1 - x * exp(4*x)).

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A346893 Expansion of e.g.f. 1 / (1 - x^5 * exp(x) / 5!).

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A358081 Expansion of e.g.f. 1/(1 - x^3 * exp(x)).

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A364981 E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x*A(x)^3).

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A380050 E.g.f. A(x) satisfies A(x) = sqrt( 1 + 2*x*exp(x)*A(x) ).

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A364981 E.g.f. satisfies A(x) = 1 + xA(x)exp(x*A(x)^3).

A380050 E.g.f. A(x) satisfies A(x) = sqrt( 1 + 2xexp(x)*A(x) ).

A380051 E.g.f. A(x) satisfies A(x) = ( 1 + 3xexp(x)*A(x) )^(1/3).