A088957 -id:A088957 - cp's OEIS frontend

A372321 Expansion of e.g.f. -exp( x + LambertW(-3*x)/3 ).

-1, 0, 6, 81, 1620, 45765, 1671678, 74794671, 3958829640, 241898775273, 16756621904970, 1297547591499819, 111065107263415308, 10412999996499836541, 1061234184094567585326, 116812280111404106348415, 13810631408232372091755792, 1745470697932523785587735249

Offset: 0

Seiichi Manyama, Apr 27 2024

sign

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A088957, A360193, A372315, A372316, A372320.

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

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

a(n) = Sum_{k=0..n} (3*k-1)^(k-1) * binomial(n,k).
G.f.: Sum_{k>=0} (3*k-1)^(k-1) * x^k / (1-x)^(k+1).
a(n) ~ 3^(n-1) * n^(n-1) * exp((exp(-1) - 1)/3). - Vaclav Kotesovec, May 06 2024

A368602 Triangle read by rows where T(n,k) is the number of labeled acyclic digraphs on {1..n} with sinks {1..k}.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 5, 3, 1, 0, 79, 33, 7, 1, 0, 3377, 1071, 161, 15, 1, 0, 362431, 92289, 10591, 705, 31, 1, 0, 93473345, 19856703, 1832705, 93375, 2945, 63, 1, 0, 56272471039, 10249747713, 789619327, 32382465, 782719, 12033, 127, 1

Offset: 0

Gus Wiseman, Jan 02 2024

Also the number of set-systems with n vertices and n edges such that {i} is a singleton edge iff i <= k, and such that there is only one way to choose a different vertex from each edge.

			Triangle begins:
    1
    0    1
    0    1    1
    0    5    3    1
    0   79   33    7    1
    0 3377 1071  161   15    1
    ...
Row n = 3 counts the following set-systems:
  {{1},{1,2},{1,3}}    {{1},{2},{1,3}}    {{1},{2},{3}}
  {{1},{1,2},{2,3}}    {{1},{2},{2,3}}
  {{1},{1,3},{2,3}}    {{1},{2},{1,2,3}}
  {{1},{1,2},{1,2,3}}
  {{1},{1,3},{1,2,3}}

Column k = n-1 is A000225 = A058877(n)/n.
Column k = 1 is A134531 (up to sign) or A003025(n)/n, non-fixed A350415.
For any choice of k sinks we get A361718.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems, unlabeled A323819.
Cf. A000169, A003024, A003087, A082402, A088957, A334282, A367862, A367904, A367908, A368600, A368601.

Table[Length[Select[Subsets[Subsets[Range[n]],{n}], Union@@Cases[#,{_}]==Range[k] && Length[Select[Tuples[#],UnsameQ@@#&]]==1&]], {n,0,3},{k,0,n}]

T(n,k) = A361718(n,k)/binomial(n,k).

More terms from Alois P. Heinz, Jan 04 2024

A088659 a(n) = n*(p-1) where p is the largest prime factor of n.

Original entry on oeis.org

2, 6, 4, 20, 12, 42, 8, 18, 40, 110, 24, 156, 84, 60, 16, 272, 36, 342, 80, 126, 220, 506, 48, 100, 312, 54, 168, 812, 120, 930, 32, 330, 544, 210, 72, 1332, 684, 468, 160, 1640, 252, 1806, 440, 180, 1012, 2162, 96, 294, 200, 816, 624, 2756, 108, 550, 336, 1026

Offset: 2

Benoit Cloitre, Nov 21 2003

nonn

It is conjectured that sequence gives period length of the periodic sequence {A088957(k) mod n}_{k>n}.

The records of this sequence are given by A036689 (product of a prime and the previous number). - Michel Marcus, May 19 2015

Ivan Neretin, Table of n, a(n) for n = 2..1000

seq(n*(max(numtheory:-factorset(n))-1), n=2..100); # Robert Israel, May 19 2015

Table[n*(FactorInteger[n][[-1, 1]] - 1), {n, 2, 57}] (* Ivan Neretin, May 19 2015 *)

PARI
```
a(n)=n*(component(factor(n),1)-1)
```

For p the k-th prime, a(p) = A036689(k). - Michel Marcus, May 19 2015

a(n) = n*A070777(n). - Michel Marcus, May 19 2015

A203092 Triangular array read by rows. T(n,k) is the number of partial functions on {1,2,...,n} that are endofunctions with no cycles of length > 1 that have exactly k components.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 18, 9, 1, 1, 116, 78, 16, 1, 1, 1060, 810, 220, 25, 1, 1, 12702, 10335, 3260, 495, 36, 1, 1, 187810, 158613, 54740, 9835, 966, 49, 1, 1, 3296120, 2854908, 1046024, 209510, 24696, 1708, 64, 1

Offset: 0

Geoffrey Critzer, Dec 29 2011

Row sums = A088957.
T(n,0)= 1,  the empty function.
T(n,n)= 1,  the identity function.
T(n,n-1)= n^2  (apparently).

			T(2,1)= 4 because there are 4 such partial functions on {1,2}: 1->1, 2->2, 1->1 2->1, 1->2 2->2,
1
1     1
1     4     1
1     18    9     1
1     116   78    16    1
1     1060  810   220   25    1
1     12702 10335 3260  495   36    1

Cf. A088956, A144289

nn = 8; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}];
f[list_] := Select[list, # > 0 &];
Map[f, Range[0, nn]! CoefficientList[ Series[Exp[x] Exp[y t], {x, 0, nn}], {x, y}]] // Flatten

E.g.f.: exp(x)*exp(y T(x)) where T(x) is the e.g.f. for A000169.

A362524 a(n) = n! * Sum_{k=0..floor(n/2)} (k+1)^(k-1) / (2^k * k! * (n-2*k)!).

Original entry on oeis.org

1, 1, 2, 4, 16, 56, 391, 2017, 20504, 139456, 1867681, 15751451, 262263442, 2638794094, 52589415971, 614628436801, 14274125637256, 190012483804952, 5041005195499849, 75288391385094811, 2246914521052963166, 37204717212894726706, 1233884675800841217847

Offset: 0

Seiichi Manyama, Apr 23 2023

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A088957, A362525.

Cf. A362350, A362522.

Table[n!Sum[(k+1)^(k-1)/(2^k k!(n-2k)!),{k,0,Floor[n/2]}],{n,0,25}] (* Harvey P. Dale, Mar 30 2024 *)

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

E.g.f.: exp(x - LambertW(-x^2/2)) = -2 * LambertW(-x^2/2)/x^2 * exp(x).

a(n) ~ (exp(sqrt(2)*exp(-1/2) + 1) + (-1)^n*exp(1 - sqrt(2)*exp(-1/2))) * n^(n-1) / (2^((n-1)/2) * exp(n/2)). - Vaclav Kotesovec, Aug 05 2025

A362525 a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)^(k-1) / (6^k * k! * (n-3*k)!).

Original entry on oeis.org

1, 1, 1, 2, 5, 11, 51, 246, 897, 7085, 51221, 260426, 2938541, 28279967, 184234415, 2714662406, 32614422401, 259026339161, 4721237878537, 67998862785970, 637019875964341, 13852253151455251, 232584488748665131, 2510358957337412182, 63466995535914172225

Offset: 0

Seiichi Manyama, Apr 23 2023

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A088957, A362524.

Cf. A362351, A362523.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-x^3/6))))

E.g.f.: exp(x - LambertW(-x^3/6)) = -6 * LambertW(-x^3/6)/x^3 * exp(x).

cp's OEIS Frontend

A372321 Expansion of e.g.f. -exp( x + LambertW(-3*x)/3 ).

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A368602 Triangle read by rows where T(n,k) is the number of labeled acyclic digraphs on {1..n} with sinks {1..k}.

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A088659 a(n) = n*(p-1) where p is the largest prime factor of n.

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A203092 Triangular array read by rows. T(n,k) is the number of partial functions on {1,2,...,n} that are endofunctions with no cycles of length > 1 that have exactly k components.

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A362524 a(n) = n! * Sum_{k=0..floor(n/2)} (k+1)^(k-1) / (2^k * k! * (n-2*k)!).

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A362525 a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)^(k-1) / (6^k * k! * (n-3*k)!).

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