A265949 -id:A265949 - 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

A292190 Sum of n-th powers of products of terms in all partitions of n into distinct parts.

Original entry on oeis.org

1, 1, 4, 35, 337, 11925, 371081, 49032439, 3545396034, 3416952655320, 749189363202730, 598250899004413536, 2383502427069445040595, 1729793152213690218766715, 131751643363739706679145099315, 271212858254426215135033141804302

Offset: 0

Seiichi Manyama, Sep 11 2017

nonn

			5 = 4 + 1 = 3 + 2. So a(5) = 5^5 + (4*1)^5 + (3*2)^5 = 11925.

Alois P. Heinz, Table of n, a(n) for n = 0..98

Main diagonal of A292189.

Cf. A265949, A292167, A292194, A292305, A292306.

b:= proc(n, i, k) option remember; (m->
      `if`(mn, 0, i^k*b(n-i, i-1, k)))))(i*(i+1)/2)
    end:
a:= n-> b(n$3):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 11 2017

nmax = 15; Table[SeriesCoefficient[Product[(1 + k^n*x^k), {k, 1, nmax}], {x, 0, n}], {n, 0, nmax}] (* Vaclav Kotesovec, Sep 12 2017 *)

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

a(n) = [x^n] Product_{k=1..n} (1 + k^n*x^k).

A292305 a(n) = [x^n] Product_{k>=1} (1 + n^k*x^k).

Original entry on oeis.org

1, 1, 4, 54, 512, 9375, 186624, 4117715, 100663296, 3099363912, 100000000000, 3423740047332, 133741506723840, 5451751918660554, 244464150162276352, 11823135040283203125, 590295810358705651712, 31435129951680797038726, 1809934771463640728469504

Offset: 0

Vaclav Kotesovec, Sep 14 2017

nonn

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

Cf. A265949, A292190, A291698, A292306.

nmax = 20; Table[SeriesCoefficient[Product[(1+n^k*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
Flatten[{1, Table[n^n*PartitionsQ[n], {n, 1, 20}]}]

a(n) = n^n * A000009(n).

a(n) ~ exp(Pi*sqrt(n/3)) * n^(n-3/4) / (4*3^(1/4)).

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

A292306 a(n) = [x^n] Product_{k>=1} (1 + n^n*x^k).

Original entry on oeis.org

1, 1, 4, 756, 65792, 19534375, 101564310279744, 558547898753326097, 9444733810164237336576, 174449211609498720646587480, 10000000004000000000400000000010000000000, 6626407607852766876000106671521201448502431912

Offset: 0

Vaclav Kotesovec, Sep 14 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..56

Cf. A265949, A292190, A292305.

nmax = 14; Table[SeriesCoefficient[Product[(1+n^n*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]

Conjecture: log(a(n)) ~ (sqrt(2)*n^(3/2) - n/2)*log(n). - Vaclav Kotesovec, Aug 22 2018

A292311 Expansion of Product_{k>=1} 1/(1 + k^k*x^k).

Original entry on oeis.org

1, -1, -3, -24, -216, -2801, -42166, -762397, -15685040, -366477168, -9523974486, -273453483050, -8583651341879, -292700900034984, -10770969729108326, -425541512224476567, -17964544188354355022, -807097409926675847400

Offset: 0

Seiichi Manyama, Sep 14 2017

sign

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

Cf. A023882, A265949.

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

Convolution inverse of A265949.

a(n) ~ -n^n * (1 - exp(-1)/n - (exp(-1)/2 + 3*exp(-2))/n^2). - Vaclav Kotesovec, Sep 14 2017

A292406 Expansion of Product_{k>=1} ((1 + k^kx^k)/(1 - k^kx^k)).

Original entry on oeis.org

1, 2, 10, 72, 670, 7896, 113572, 1939028, 38463550, 869985586, 22098989952, 622728621984, 19271496576612, 649553583740576, 23680212403186584, 928276782505698920, 38931911577966732814, 1739307919812511213916, 82457732209611432170734

Offset: 0

Seiichi Manyama, Sep 15 2017

nonn

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

Cf. A023882, A265949, A292407.

nmax = 20; CoefficientList[Series[Product[(1 + k^k*x^k)/(1 - k^k*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 16 2017 *)

Convolution of A023882 and A265949.

a(n) ~ 2*n^n * (1 + 2*exp(-1)/n + (exp(-1) + 10*exp(-2))/n^2). - Vaclav Kotesovec, Sep 16 2017

A299786 Expansion of Product_{k>=1} (1 + k^(k-1)*x^k).

Original entry on oeis.org

1, 1, 2, 11, 73, 707, 8547, 127379, 2237804, 45511484, 1049155214, 27060763974, 771662014455, 24109614539775, 818906748562249, 30044648617150066, 1184045057676213763, 49883902402848781573, 2237286132689496359239, 106426356238092171308928, 5352031894869594850387969

Offset: 0

Ilya Gutkovskiy, Jan 21 2019

nonn

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1, g(n) = (-1) * n^(n-1). - Seiichi Manyama, Aug 22 2020

Seiichi Manyama, Table of n, a(n) for n = 0..387

Cf. A265949, A266964, A304961, A321387, A323634.

nmax = 20; CoefficientList[Series[Product[(1 + k^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^(k - k/d + 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 20}]

N=40; x='x+O('x^N); Vec(prod(k=1, N, 1+k^(k-1)*x^k)) \\ Seiichi Manyama, Aug 22 2020

a(n) ~ n^(n-1) * (1 + exp(-1)/n + (2*exp(-2) + 3*exp(-1)/2)/n^2). - Vaclav Kotesovec, Jan 22 2019

cp's OEIS Frontend

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

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A292190 Sum of n-th powers of products of terms in all partitions of n into distinct parts.

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A292305 a(n) = [x^n] Product_{k>=1} (1 + n^k*x^k).

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

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A292306 a(n) = [x^n] Product_{k>=1} (1 + n^n*x^k).

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A292311 Expansion of Product_{k>=1} 1/(1 + k^k*x^k).

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A292406 Expansion of Product_{k>=1} ((1 + k^k*x^k)/(1 - k^k*x^k)).

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A299786 Expansion of Product_{k>=1} (1 + k^(k-1)*x^k).

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A292406 Expansion of Product_{k>=1} ((1 + k^kx^k)/(1 - k^kx^k)).