A294645 -id:A294645 - cp's OEIS frontend

A023882 Expansion of g.f.: 1/Product_{n>0} (1 - n^n * x^n).

1, 1, 5, 32, 304, 3537, 52010, 895397, 18016416, 410889848, 10523505770, 298220329546, 9274349837081, 313761671751672, 11474635626789410, 450964042480390679, 18954785687060988578, 848386888530723146912

Offset: 0

Olivier Gérard

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..380

Cf. A265949, A292312, A294645.

m:=20; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-k^k*x^k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018

seq(coeff(series(1/mul(1-k^k*x^k,k=1..n),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 31 2018

nmax=20; CoefficientList[Series[Product[1/(1-k^k*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 19 2015 *)

m=20; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-k^k*x^k))) \\ G. C. Greubel, Oct 30 2018

Log of g.f.: Sum_{k>=1} (sigma(k, k+1)/k) x^k, where sigma(k, q) is the sum of the q-th powers of the divisors of k.
a(n) ~ n^n * (1 + exp(-1)/n + (1/2*exp(-1)+5*exp(-2))/n^2). - Vaclav Kotesovec, Dec 19 2015
a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} A294645(k)*a(n-k) for n > 0. - Seiichi Manyama, Nov 09 2017

A292312 Expansion of Product_{k>=1} (1 - k^k*x^k).

Original entry on oeis.org

1, -1, -4, -23, -229, -2761, -42615, -758499, -15702086, -365588036, -9516954786, -273061566624, -8575969258607, -292418459301779, -10762887030763337, -425243370397722674, -17953905924215881215, -806666656048846472309

Offset: 0

Seiichi Manyama, Sep 14 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..386

Column k=1 of A294653.

Cf. A023882, A265949.

m:=20; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1 - k^k*x^k): k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

seq(coeff(series(mul((1-k^k*x^k),k=1..n),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 31 2018

terms = 18; CoefficientList[Product[(1 - k^k*x^k), {k, 1, terms}] + O[x]^(terms), x] (* Jean-François Alcover, Nov 11 2017 *)

{a(n) = polcoeff(prod(k=1, n, 1-k^k*x^k+x*O(x^n)), n)}

Convolution inverse of A023882.
a(n) ~ -n^n * (1 - exp(-1)/n - (exp(-1)/2 + 4*exp(-2))/n^2). - Vaclav Kotesovec, Sep 14 2017
a(0) = 1 and a(n) = -(1/n) * Sum_{k=1..n} A294645(k)*a(n-k) for n > 0. - Seiichi Manyama, Nov 09 2017

A294810 a(n) = Sum_{d|n} d^(n+2).

Original entry on oeis.org

1, 17, 244, 4161, 78126, 1686434, 40353608, 1074791425, 31381236757, 1000244144722, 34522712143932, 1283997101947770, 51185893014090758, 2177986570740006274, 98526126098761952664, 4722384497336874434561, 239072435685151324847154

Offset: 1

Seiichi Manyama, Nov 09 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..384

Cf. A294645, A294809.

Table[Total[Divisors[n]^(n+2)],{n,20}] (* Harvey P. Dale, Dec 23 2023 *)

PARI
```
{a(n) = sigma(n, n+2)}
```

N=66; x='x+O('x^N); Vec(sum(k=1, N, k^(k+2)*x^k/(1-(k*x)^k)))

N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-(k*x)^k)^k)))) \\ Seiichi Manyama, Jun 02 2019

G.f.: Sum_{k>0} k^(k+2)*x^k/(1-(k*x)^k).

L.g.f.: -log(Product_{k>=1} (1 - (k*x)^k)^k) = Sum_{k>=1} a(k)*x^k/k. - Seiichi Manyama, Jun 02 2019

A308504 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} d^(n+k).

Original entry on oeis.org

1, 1, 5, 1, 9, 28, 1, 17, 82, 273, 1, 33, 244, 1057, 3126, 1, 65, 730, 4161, 15626, 47450, 1, 129, 2188, 16513, 78126, 282252, 823544, 1, 257, 6562, 65793, 390626, 1686434, 5764802, 16843009, 1, 513, 19684, 262657, 1953126, 10097892, 40353608, 134480385, 387440173

Offset: 1

Seiichi Manyama, Jun 02 2019

			a(4) = a(2*3/2 + 1) = sigma_3(1) = 1.
a(5) = a(2*3/2 + 2) = sigma_3(2) = 1^3 + 2^3 = 9.
a(6) = a(2*3/2 + 3) = sigma_3(3) = 1^3 + 3^3 = 28.
Square array begins:
       1,      1,       1,        1,        1, ...
       5,      9,      17,       33,       65, ...
      28,     82,     244,      730,     2188, ...
     273,   1057,    4161,    16513,    65793, ...
    3126,  15626,   78126,   390626,  1953126, ...
   47450, 282252, 1686434, 10097892, 60526250, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened
Index entries for sequences related to sigma(n)

Columns k=0..2 give A023887, A294645, A294810.
A(n,n) gives A308570.
Cf. A279394, A308502.

T[n_, k_] := DivisorSum[n, #^(n+k) &]; Table[T[k, n - k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, May 11 2021 *)

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

a((i-1)*i/2 + j) = sigma_i(j) for 1 <= j <= i.

A342675 a(n) = Sum_{d|n} d^(n-d+1).

Original entry on oeis.org

1, 3, 4, 13, 6, 120, 8, 1161, 2197, 16148, 12, 603190, 14, 5773008, 50422464, 201359377, 18, 16590656229, 20, 269768284118, 4748723771432, 3138430473896, 24, 2972582195034162, 476837158203151, 3937376419253748, 1350852564961601560, 4066515044181860654, 30, 1036488835382356683530, 32

Offset: 1

Seiichi Manyama, Mar 18 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..599

Cf. A023887, A294645, A342628, A342677.

a[n_] := DivisorSum[n, #^(n - # + 1) &]; Array[a, 30] (* Amiram Eldar, Mar 18 2021 *)

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

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, k*x^k/(1-(k*x)^k)))

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

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

A158095 G.f.: A(x) = exp( Sum_{n>=1} 2sigma(n,n-1)x^n/n ).

Original entry on oeis.org

1, 2, 5, 14, 61, 370, 3454, 40880, 614346, 10848514, 222870183, 5175100204, 134514302384, 3859406052466, 121274242936381, 4139268759072626, 152532132931199062, 6034430112251517608, 255114747410233804986

Offset: 0

Paul D. Hanna, Mar 20 2009

nonn

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 14*x^3 + 61*x^4 + 370*x^5 +...

Cf. A023881, A294645.

a(n)=polcoeff(exp(sum(m=1,n,2*sigma(m,m-1)*x^m/m)+x*O(x^n)),n)

A294773 a(n) = Sum_{d|n} d^(d+n+1).

Original entry on oeis.org

1, 33, 2188, 262273, 48828126, 13060753578, 4747561509944, 2251799880796161, 1350851717674586413, 1000000000152587898818, 895430243255237372246532, 953962166441299506564257602, 1192533292512492016559195008118

Offset: 1

Seiichi Manyama, Nov 08 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..214

Cf. A294645, A294704, A294757.

a[n_] := DivisorSum[n, #^(# + n + 1) &]; Array[a, 13] (* Amiram Eldar, Oct 04 2023 *)

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

A294955 a(n) = Sum_{d|n} d^(2*n+2).

Original entry on oeis.org

1, 65, 6562, 1049601, 244140626, 78368963450, 33232930569602, 18014467229220865, 12157665462543713203, 10000002384185795209930, 9849732675807611094711842, 11447546167874515876354097130, 15502932802662396215269535105522

Offset: 1

Seiichi Manyama, Nov 12 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..213

Cf. A294645, A294810, A294953, A294954.

PARI
```
{a(n) = sigma(n, 2*n+2)}
```

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

G.f.: Sum_{k>0} k^(2*k+2)*x^k/(1-(k^2*x)^k).

A342677 a(n) = Sum_{d|n} (n/d)^(n-d+1).

Original entry on oeis.org

1, 5, 28, 265, 3126, 46916, 823544, 16793633, 387422677, 10001953190, 285311670612, 8916464313700, 302875106592254, 11112103714568680, 437893891601739648, 18446779258148749825, 827240261886336764178, 39346424755299348744797, 1978419655660313589123980

Offset: 1

Seiichi Manyama, Mar 18 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A294645, A342629, A342675.

a[n_] := DivisorSum[n, (n/#)^(n - # + 1) &]; Array[a, 20] (* Amiram Eldar, Mar 18 2021 *)

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

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

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

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

A308763 a(n) = Sum_{d|n} d^(n-2).

Original entry on oeis.org

1, 2, 4, 21, 126, 1394, 16808, 266305, 4785157, 100390882, 2357947692, 61978939050, 1792160394038, 56707753666594, 1946196290656824, 72061992352890881, 2862423051509815794, 121441386937936123331, 5480386857784802185940, 262145000003883417004506

Offset: 1

Seiichi Manyama, Jun 23 2019

nonn

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

Cf. A023887, A082245, A294645, A294810, A308755.

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

PARI
```
{a(n) = sigma(n, n-2)}
```

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

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

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

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

cp's OEIS Frontend

A023882 Expansion of g.f.: 1/Product_{n>0} (1 - n^n * x^n).

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A292312 Expansion of Product_{k>=1} (1 - k^k*x^k).

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A294810 a(n) = Sum_{d|n} d^(n+2).

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A308504 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} d^(n+k).

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A342675 a(n) = Sum_{d|n} d^(n-d+1).

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A158095 G.f.: A(x) = exp( Sum_{n>=1} 2*sigma(n,n-1)*x^n/n ).

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A294773 a(n) = Sum_{d|n} d^(d+n+1).

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Mathematica

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A294955 a(n) = Sum_{d|n} d^(2*n+2).

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A158095 G.f.: A(x) = exp( Sum_{n>=1} 2sigma(n,n-1)x^n/n ).