A262843 -id:A262843 - cp's OEIS frontend

A308814 a(n) = Sum_{d|n} n^(d-1).

1, 3, 10, 69, 626, 7819, 117650, 2097673, 43046803, 1000010011, 25937424602, 743008621405, 23298085122482, 793714780783695, 29192926025441476, 1152921504875286545, 48661191875666868482, 2185911559749718382455, 104127350297911241532842, 5242880000000512000168021

Offset: 1

Seiichi Manyama, Jun 26 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..387

Cf. A066108, A082245, A262843, A308813.

Table[Total[n^(Divisors[n]-1)],{n,20}] (* Harvey P. Dale, Aug 08 2019 *)

PARI
```
{a(n) = sumdiv(n, d, n^(d-1))}
```

a(n) = A308813(n,n).
a(n) = A066108(n)/n.
a(n) ~ n^(n-1). - Vaclav Kotesovec, Jun 05 2021

A320444 Number of uniform hypertrees spanning n vertices.

Original entry on oeis.org

1, 1, 1, 4, 17, 141, 1297, 17683, 262145, 4861405, 100112001, 2371816701, 61917364225, 1796326510993, 56693912375297, 1947734359001551, 72059082110369793, 2863257607266475419, 121439531096594251777, 5480987217944109919765, 262144000000000000000001

Offset: 0

Gus Wiseman, Jan 09 2019

nonn

The density of a hypergraph is the sum of sizes of its edges minus the number of edges minus the number of vertices. A hypertree is a connected hypergraph of density -1. A hypergraph is uniform if its edges all have the same size. The span of a hypergraph is the union of its edges.

			Non-isomorphic representatives of the 5 unlabeled uniform hypertrees on 5 vertices and their multiplicities in the labeled case, which add up to a(5) = 141:
   5 X {{1,5},{2,5},{3,5},{4,5}}
  60 X {{1,4},{2,5},{3,5},{4,5}}
  60 X {{1,3},{2,4},{3,5},{4,5}}
  15 X {{1,2,5},{3,4,5}}
   1 X {{1,2,3,4,5}}

Robert Israel, Table of n, a(n) for n = 0..387

Row sums of A326374.

Cf. A000272, A030019, A035053, A038041, A052888, A057625, A061095, A121860, A134954, A236696, A262843.

f:= proc(n) local d; add((n-1)!/(d! * ((n-1)/d)!) * (n/d)^((n-1)/d - 1), d = numtheory:-divisors(n-1)); end proc:
f(0):= 1: f(1):= 1:
map(f, [$0..25]); # Robert Israel, Jan 10 2019

Table[Sum[n!/(d!*(n/d)!)*((n+1)/d)^(n/d-1),{d,Divisors[n]}],{n,10}]

a(n) = if (n<2, 1, n--; sumdiv(n, d, n!/(d! * (n/d)!) * ((n + 1)/d)^(n/d - 1))); \\ Michel Marcus, Jan 10 2019

a(n + 1) = Sum_{d|n} n!/(d! * (n/d)!) * ((n + 1)/d)^(n/d - 1).

a(p prime) = 1 + (p + 1)^(p - 1).

A308701 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*(d-1)).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 5, 10, 3, 1, 9, 82, 67, 2, 1, 17, 730, 4101, 626, 4, 1, 33, 6562, 262153, 390626, 7788, 2, 1, 65, 59050, 16777233, 244140626, 60466262, 117650, 4, 1, 129, 531442, 1073741857, 152587890626, 470184985314, 13841287202, 2097219, 3

Offset: 1

Seiichi Manyama, Jun 22 2019

			Square array begins:
   1,   1,      1,         1,            1, ...
   2,   3,      5,         9,           17, ...
   2,  10,     82,       730,         6562, ...
   3,  67,   4101,    262153,     16777233, ...
   2, 626, 390626, 244140626, 152587890626, ...

Seiichi Manyama, Antidiagonals n = 1..53, flattened

Columns k=0..2 give A000005, A262843, A308753.
Row n=1..3 give A000012, A000051, A062396.
Cf. A308694, A308698, A308704.

T[n_, k_] := DivisorSum[n, #^(k*(# - 1)) &]; Table[T[k, n - k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, May 09 2021 *)

L.g.f. of column k: -log(Product_{j>=1} (1 - x^j)^(j^(k*j-k-1))).

G.f. of column k: Sum_{j>=1} j^(k*(j-1)) * x^j/(1 - x^j).

A308755 a(n) = Sum_{d|n} d^(d-2).

Original entry on oeis.org

1, 2, 4, 18, 126, 1301, 16808, 262162, 4782973, 100000127, 2357947692, 61917365541, 1792160394038, 56693912392105, 1946195068359504, 72057594038190098, 2862423051509815794, 121439531096599036046, 5480386857784802185940, 262144000000000100000143

Offset: 1

Seiichi Manyama, Jun 22 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..388

Cf. A062796, A262843, A283498, A308753.

a[n_] := DivisorSum[n, #^(# - 2) &]; Array[a, 20] (* Amiram Eldar, May 08 2021 *)

PARI
```
{a(n) = sumdiv(n, d, d^(d-2))}
```

N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-x^k)^k^(k-3)))))

N=20; x='x+O('x^N); Vec(sum(k=1, N, k^(k-2)*x^k/(1-x^k)))

L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^(k-3))) = Sum_{k>=1} a(k)*x^k/k.

G.f.: Sum_{k>=1} k^(k-2) * x^k/(1 - x^k).

A321386 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^(d-1).

Original entry on oeis.org

1, 1, 10, 61, 626, 7768, 117650, 2097085, 43046731, 999999376, 25937424602, 743008362964, 23298085122482, 793714773136496, 29192926025391260, 1152921504604749757, 48661191875666868482, 2185911559738653493015, 104127350297911241532842, 5242879999999998999999436, 278218429446951548637314060

Offset: 1

Ilya Gutkovskiy, Nov 08 2018

nonn

Cf. A000169, A262843, A321388.

Table[Sum[(-1)^(n/d + 1) d^(d - 1), {d, Divisors[n]}], {n, 21}]
nmax = 21; Rest[CoefficientList[Series[Sum[k^(k - 1) x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]

a(n) = sumdiv(n, d, (-1)^(n/d+1)*d^(d-1)); \\ Michel Marcus, Nov 09 2018

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

a(n) ~ n^(n-1). - Vaclav Kotesovec, Nov 09 2018

A383003 a(n) = Sum_{d|n} (-n)^(d-1).

Original entry on oeis.org

1, -1, 10, -67, 626, -7745, 117650, -2097671, 43046803, -999990009, 25937424602, -743008621115, 23298085122482, -793714765724621, 29192926025441476, -1152921504875286543, 48661191875666868482, -2185911559727678460653, 104127350297911241532842

Offset: 1

Seiichi Manyama, Apr 12 2025

Cf. A101561, A101562, A101563.
Cf. A262843, A308814, A383010.
Cf. A382998, A383012.

PARI
```
a(n) = sumdiv(n, d, (-n)^(d-1));
```

a(n) = (1/n) * A383010(n).
a(n) = [x^n] Sum_{k>=1} log(1 + n*x^k) / k.
a(n) = [x^n] Sum_{k>=1} x^k / (1 + n*x^k).

A262842 G.f.: Product_{k>=1} (1 - x^k)^(-k^(k-2)).

Original entry on oeis.org

1, 1, 2, 5, 22, 150, 1469, 18452, 282426, 5088276, 105431378, 2469403421, 64508609896, 1859464257187, 58625171707730, 2006861834895431, 74128128916520263, 2938711927441481562, 124457492116819509679, 5607967808192795374759, 267883606645817181302028, 13522287374792657280601627, 719232962773594118661491002, 40204966834965724305054746851

Offset: 0

Paul D. Hanna, Oct 03 2015

nonn

			 G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 22*x^4 + 150*x^5 + 1469*x^6 +...
where
A(x) = 1/((1-x)*(1-x^2)*(1-x^3)^3*(1-x^4)^16*(1-x^5)^125*(1-x^6)^1296*...)
also
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 67*x^4/4 + 626*x^5/5 + 7788*x^6/6 + 117650*x^7/7 + 2097219*x^8/8 + 43046731*x^9/9 + 1000000628*x^10/10 +...+ A262843(n)*x^n/n +...

Cf. A262843, A023879.

{a(n)=polcoeff(prod(k=1, n, (1 - x^k +x*O(x^n))^(-k^(k-2))), n)}
for(n=0,30,print1(a(n),", "))

{a(n)=polcoeff(exp(sum(m=1, n+1, sumdiv(m, d, d^d)*x^m/m) +x*O(x^n)), n)}
for(n=0,30,print1(a(n),", "))

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{d|n} d^(d-1) ).

Logarithmic derivative equals A262843, where A262843(n) = Sum_{d|n} d^(d-1).

A326374 Irregular triangle read by rows where T(n,k) is the number of (d + 1)-uniform hypertrees spanning n + 1 vertices, where d = A027750(n,k).

Original entry on oeis.org

1, 3, 1, 16, 1, 125, 15, 1, 1296, 1, 16807, 735, 140, 1, 262144, 1, 4782969, 76545, 1890, 1, 100000000, 112000, 1, 2357947691, 13835745, 33264, 1, 61917364224, 1, 1792160394037, 3859590735, 270670400, 35135100, 720720, 1, 56693912375296, 1, 1946195068359375

Offset: 1

Gus Wiseman, Jul 03 2019

A hypertree is a connected hypergraph of density -1, where density is the sum of sizes of the edges minus the number of edges minus the number of vertices. A hypergraph is k-uniform if its edges all have size k. The span of a hypertree is the union of its edges.

			Triangle begins:
           1
           3          1
          16          1
         125         15          1
        1296          1
       16807        735        140          1
      262144          1
     4782969      76545       1890          1
   100000000     112000          1
  2357947691   13835745      33264          1
The T(4,2) = 15 hypertrees:
  {{1,4,5},{2,3,5}}
  {{1,4,5},{2,3,4}}
  {{1,3,5},{2,4,5}}
  {{1,3,5},{2,3,4}}
  {{1,3,4},{2,4,5}}
  {{1,3,4},{2,3,5}}
  {{1,2,5},{3,4,5}}
  {{1,2,5},{2,3,4}}
  {{1,2,5},{1,3,4}}
  {{1,2,4},{3,4,5}}
  {{1,2,4},{2,3,5}}
  {{1,2,4},{1,3,5}}
  {{1,2,3},{3,4,5}}
  {{1,2,3},{2,4,5}}
  {{1,2,3},{1,4,5}}

Alois P. Heinz, Rows n = 1..185, flattened

Row sums are A320444.
Column d = 1 is A000272.
Cf. A030019, A027750, A035053, A038041, A052888, A057625, A061095, A121860, A134954, A236696, A262843.

T:= n-> seq(n!/(d!*(n/d)!)*((n+1)/d)^(n/d-1), d=numtheory[divisors](n)):
seq(T(n), n=1..20);  # Alois P. Heinz, Aug 21 2019

Table[n!/(d!*(n/d)!)*((n+1)/d)^(n/d-1),{n,10},{d,Divisors[n]}]

T(n, k) = n!/(d! * (n/d)!) * ((n + 1)/d)^(n/d - 1), where d = A027750(n, k).

Edited by Peter Munn, Mar 05 2025

A345094 a(n) = Sum_{k=1..n} floor(n/k)^(floor(n/k) - 1).

Original entry on oeis.org

1, 3, 11, 68, 630, 7790, 117664, 2097224, 43046801, 1000000643, 25937425245, 743008378547, 23298085130341, 793714773371879, 29192926025508929, 1152921504608944840, 48661191875668966346, 2185911559738739586562, 104127350297911284587436

Offset: 1

Seiichi Manyama, Jun 07 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..387

Cf. A060946, A262843, A344551, A345030, A345098.

a[n_] := Sum[Floor[n/k]^(Floor[n/k] - 1), {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Jun 08 2021 *)

PARI
```
a(n) = sum(k=1, n, (n\k)^(n\k-1));
```

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

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

A354889 a(n) = n! * Sum_{d|n} d^(d-1) / d!.

Original entry on oeis.org

1, 4, 15, 112, 745, 10296, 122689, 2285312, 43953921, 1026157600, 25977341401, 751135431168, 23304312143281, 795924137531264, 29203006015310625, 1154107395053387776, 48661547563094964481, 2186762596692631699968, 104127471943011650364841

Offset: 1

Seiichi Manyama, Jun 10 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..387

Cf. A262843, A327578, A354845, A354888.

a[n_] := n! * DivisorSum[n, #^(# - 1)/#! &]; Array[a, 19] (* Amiram Eldar, Jun 10 2022 *)

PARI
```
a(n) = n!*sumdiv(n, d, d^(d-1)/d!);
```

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

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

If p is prime, a(p) = p^(p-1) + p!.

cp's OEIS Frontend

A308814 a(n) = Sum_{d|n} n^(d-1).

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A320444 Number of uniform hypertrees spanning n vertices.

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A308701 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*(d-1)).

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A308755 a(n) = Sum_{d|n} d^(d-2).

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PARI

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A321386 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^(d-1).

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Mathematica

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A383003 a(n) = Sum_{d|n} (-n)^(d-1).

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A262842 G.f.: Product_{k>=1} (1 - x^k)^(-k^(k-2)).

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PARI

PARI

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A326374 Irregular triangle read by rows where T(n,k) is the number of (d + 1)-uniform hypertrees spanning n + 1 vertices, where d = A027750(n,k).

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