A168243 -id:A168243 - cp's OEIS frontend

A028342 Expansion of Product_{i>=1} (1 - x^i)^(-1/i); also of exp(Sum_{n>=1} (d(n)*x^n/n)) where d is number of divisors function.

1, 1, 3, 11, 59, 339, 2629, 20677, 202089, 2066201, 24322931, 296746251, 4193572723, 59806188571, 954679763829, 15845349818789, 285841314451409, 5293203821406897, 106976406006818659, 2201383054398314251

Offset: 0

Jeffrey Shallit

From Peter Bala, Nov 14 2017: (Start)
It appears that the sequence taken modulo 10 is periodic with period 10. More generally, we conjecture
(1) for k odd, a(n+k) + a(n) is divisible by k: if true, then for k odd, the sequence a(n) taken modulo k would be periodic with period dividing 2*k.
(2) for even k congruent to 0, 2 or 6 modulo 8 then a(n+k) - a(n) is divisible by k; in these cases the sequence a(n) taken modulo k would be periodic with period dividing k.
(3) for even k congruent to 4 modulo 8 then 2*( a(n+k) - a(n) ) is divisible by k; in this case the sequence 2*a(n) taken modulo k would be periodic with period dividing k. (End)
a(n) is the number of colored permutations by number of divisors, that is, permutations whose decomposition into a product of cycles gives the result that each cycle carries a label that is a divisor of its corresponding length. - Ricardo Gómez Aíza, Mar 08 2023

			For n = 3, there are 6 permutations that written as product of cycles are (1)(2)(3), (1)(23), (2)(13), (3)(12), (123), (132). Cycles of length one can only carry the label 1. Cycles of length two can carry the label either 1 or 2. Cycles of length three can carry the label either 1 or 3. Then a(3) = 11. - _Ricardo Gómez Aíza_, Mar 08 2023

Vaclav Kotesovec, Table of n, a(n) for n = 0..445
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.
N. J. A. Sloane, Transforms

Cf. A000005, A168243.

nmax=20; CoefficientList[Series[Product[1/(1-x^k)^(1/k),{k,1,nmax}],{x,0,nmax}],x] * Range[0,nmax]! (* Vaclav Kotesovec, May 28 2015 *)
a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSigma[0, k]*a[n-k], {k, 1, n}]/n]; Table[n!*a[n], {n, 0, 20}] (* Vaclav Kotesovec, Sep 07 2018 *)
nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[0, k]*x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 26 2019 *)

a(n):=if n=0 then 1 else (n-1)!*sum(length(divisors(i+1))*a(n-i-1)/(n-i-1)!,i,0,n-1); /* Vladimir Kruchinin, Feb 27 2015 */

This is an expansion as an exponential generating function, i.e., as sum a(n)*x^n/n!.
Equivalently, a(n)/n! is the Euler transform of [1, 1/2, 1/3, 1/4, ...].
a(n) = (n-1)!*Sum_{i=0..n-1} d(i+1)*a(n-i-1)/(n-i-1)!, a(0)=1, where d(i) is number of divisors function. - Vladimir Kruchinin, Feb 27 2015
Conjecture: log(a(n)/n!) ~ log(2)/2 * log(n)^2. - Vaclav Kotesovec, Sep 15 2018
From Ricardo Gómez Aíza, Mar 08 2023: (Start)
The above conjecture is incorrect:
a(n)/n! ~ (w(n) / n)^(1 - gamma)/sqrt(2 * Pi * abs(log(w(n) / n))) * exp(c + w(n) + (log(w(n) / n))^2 / 2), where w(n) = W(e^gamma * n), W is the Lambert W function, gamma is the Euler-Mascheroni constant, c = Pi^2 / 12 - gamma^2 / 2 - 2 * gamma(1), and gamma(1) is the 1st Stieltjes number.
log(a(n)/n!) ~ (1/2) * log(n)^2. (End)

Edited by Franklin T. Adams-Watters, Jul 03 2009

A206303 Expansion of e.g.f.: Product_{n>=1} (1 - x^(2n-1))^(-1/(2n-1)).

Original entry on oeis.org

1, 1, 2, 8, 32, 184, 1264, 9568, 79232, 816128, 8769536, 101867776, 1322831872, 18122579968, 268425347072, 4436611211264, 73309336469504, 1303024044310528, 25235367455752192, 497968598916333568, 10431118327503650816, 234674470003955204096, 5359992446798535852032

Offset: 0

Paul D. Hanna, Feb 06 2012

nonn

			G.f.: A(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 32*x^4/4! + 184*x^5/5! + ...
The e.g.f. equals the product:
A(x) = (1-x)^(-1) * (1-x^3)^(-1/3) * (1-x^5)^(-1/5) * (1-x^7)^(-1/7) * (1-x^9)^(-1/9) * (1-x^11)^(-1/11) * ...

Alois P. Heinz, Table of n, a(n) for n = 0..448

Cf. A000005, A001227, A028342, A168243.

m:=40;
f:= func< x | (&*[1/(1 - x^(2*n-1))^(1/(2*n-1)) : n in [1..m+10]]) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!(Laplace( f(x) ))); // G. C. Greubel, Dec 21 2022

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(d::odd, 1, 0), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> b(n)*n!:
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 24 2017

b[n_]:= b[n]= If[n==0, 1, Sum[Sum[If[OddQ[d], 1, 0], {d, Divisors[j]}]* b[n-j], {j, n}]/n];
a[n_]:= b[n]*n!;
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 10 2018, after Alois P. Heinz *)

{a(n)=n!*polcoeff(prod(m=1,n,(1-x^(2*m-1)+x*O(x^n))^(-1/(2*m-1))),n)}
for(n=0,31,print1(a(n),", "))

m=40
def f(x): return 1/product( (1 - x^(2*n-1))^(1/(2*n-1)) for n in range(1,m+11) )
def A206303_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).egf_to_ogf().list()
A206303_list(m+1) # G. C. Greubel, Dec 21 2022

a(n)/n! is the Euler transform of [1, 0, 1/3, 0, 1/5, 0, 1/7, 0, ...].
E.g.f.: A(x) = B(x) / sqrt(B(x^2)), where B(x) = e.g.f. of A028342.
E.g.f. A(x) satisfies: Product_{n>=0} A(x^(2^n))^(1/2^n) = e.g.f. of A028342.
E.g.f.: exp(Sum_{k>=1} (d(2*k) - d(k))*x^k/k), where d(k) = number of divisors of k (A000005). - Ilya Gutkovskiy, Sep 17 2018

A298906 Expansion of e.g.f. Product_{k>=1} (1 + x^k)^(1/k!).

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 77, 29, -4289, -14836, 283812, 1316855, -16548717, -292820579, 911200565, 52594983250, 100157634380, -3444629077653, 7961210574683, -2170805244559295, -41176659971108705, 348776485253486302, 35663019455311634058, 513993485453689440281

Offset: 0

Ilya Gutkovskiy, Jun 18 2018

sign

			E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 2*x^4/4! + x^5/5! + 77*x^6/6! + ... = (1 + x) * (1 + x^2)^(1/2!) * (1 + x^3)^(1/3!) * (1 + x^4)^(1/4!) * ...

Seiichi Manyama, Table of n, a(n) for n = 0..450

Cf. A000110, A168243, A209902, A345762.

a:=series(exp(add((-1)^(k+1)*(exp(x^k)-1)/k,k=1..100)),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 26 2019

nmax = 23; CoefficientList[Series[Exp[Sum[(-1)^(k + 1) (Exp[x^k] - 1)/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1)/(d - 1)!, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 23}]

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

E.g.f.: Product_{k>=1} B(x^k)^((-1)^(k+1)/k), where B(x) = exp(exp(x) - 1) = e.g.f. of Bell numbers (A000110).

A356392 Expansion of e.g.f. ( Product_{k>0} (1+x^k)^(1/k) )^(1/(1-x)).

Original entry on oeis.org

1, 1, 3, 17, 99, 769, 6877, 70769, 807321, 10366037, 145721531, 2226927405, 36741898267, 651709348653, 12352436747141, 249152882935829, 5320544034698353, 120008265471779529, 2850195632804141203, 71058458112629765449, 1855470903727083981651

Offset: 0

Seiichi Manyama, Aug 05 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..440

Cf. A356393, A356394.

Cf. A168243, A356336, A356389.

my(N=30, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+x^k)^(1/k))^(1/(1-x))))

a356389(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1))/k);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356389(j)*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} A356389(k) * binomial(n-1,k-1) * a(n-k).

A294356 E.g.f.: Product_{k>0} (1+x^k)^(-1/k).

Original entry on oeis.org

1, -1, 1, -5, 23, -119, 619, -4759, 48145, -476657, 4249961, -48286061, 691331431, -9132207655, 117900772963, -2025161870159, 37607411624609, -628236985455329, 10768798391659345, -215626810984559317, 4751529623277906871, -105427459848063440471

Offset: 0

Seiichi Manyama, Oct 29 2017

sign

Cf. A028342, A028343, A168243.

nmax = 20; CoefficientList[Series[Product[1/(1+x^k)^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 29 2017 *)

N=66; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, 1/(1+x^k)^(1/k))))

A295792 Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(1/k).

Original entry on oeis.org

1, 2, 6, 28, 152, 1008, 7936, 70208, 689664, 7618816, 92013824, 1202362368, 17053410304, 258928934912, 4197838491648, 72840915607552, 1334630802489344, 25799982480556032, 527187369241870336, 11292834065764450304, 253498950169144590336, 5965951790211865772032, 146341359815078034538496

Offset: 0

Ilya Gutkovskiy, Nov 27 2017

nonn

Convolution of A028342 and A168243. - Vaclav Kotesovec, Sep 07 2018

Vaclav Kotesovec, Table of n, a(n) for n = 0..445

Cf. A001227, A028342, A028343, A054844, A156616, A168243, A206303, A294356.

a:=series(mul(((1+x^k)/(1-x^k))^(1/k),k=1..100),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 27 2019

nmax = 22; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: exp(2*Sum_{k>=1} A001227(k)*x^k/k).

E.g.f.: exp(Sum_{k>=1} A054844(k)*x^k/k).

A318696 Expansion of e.g.f. Product_{i>=1, j>=1} (1 + x^(ij))^(1/(ij)).

Original entry on oeis.org

1, 1, 2, 10, 34, 218, 1708, 12556, 97340, 1139932, 12602584, 142757624, 1983086488, 26745019000, 402951386576, 7181178238672, 115410887636752, 2039658743085584, 42354537803172640, 815690033731561888, 17593347085888752416, 416765224159172991136, 9379433694333768563392

Offset: 0

Ilya Gutkovskiy, Aug 31 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..446
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.

Cf. A000005, A107742, A168243, A280541, A288571, A318695, A318977.

seq(n!*coeff(series(mul((1+x^k)^(tau(k)/k),k=1..100),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 09 2019

nmax = 22; CoefficientList[Series[Product[Product[(1 + x^(i j))^(1/(i j)), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Product[(1 + x^k)^(DivisorSigma[0, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k/d + 1) DivisorSigma[0, d], {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]
nmax = 22; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[0, k]/k, j]*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; Take[CoefficientList[s, x], nmax + 1] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 01 2018 *)

E.g.f.: Product_{k>=1} (1 + x^k)^(tau(k)/k), where tau = number of divisors (A000005).

E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*tau(d) ) * x^k/k).

A318769 Expansion of e.g.f. Product_{k>=1} (1 + x^k)^(sigma(k)/k), where sigma(k) is the sum of the divisors of k.

Original entry on oeis.org

1, 1, 3, 17, 83, 639, 5749, 53227, 561273, 7216577, 94292531, 1352253561, 21657812923, 359338829407, 6460367397093, 126124578755939, 2527688612931569, 54137820027005697, 1236730462664172643, 29137619131277727457, 725282418459957414051, 18981526480933601454911

Offset: 0

Ilya Gutkovskiy, Sep 03 2018

nonn

a(n)/n! is the weigh transform of [1, 3/2, 4/3, 7/4, 6/5, ... = sums of reciprocals of divisors of 1, 2, 3, 4, 5, ...].

Vaclav Kotesovec, Table of n, a(n) for n = 0..440
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.
N. J. A. Sloane, Transforms

Cf. A000203, A017665, A017666, A168243, A192065, A288417, A305127, A318696, A318814.

with(numtheory): a := proc(n) option remember; `if`(n = 0, 1, add(add(-(-1)^(j/d)*sigma(d), d = divisors(j))*a(n-j), j = 1..n)/n) end proc; seq(n!*a(n), n = 0..20); # Vaclav Kotesovec, Sep 04 2018

nmax = 21; CoefficientList[Series[Product[(1 + x^k)^(DivisorSigma[1, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k + 1) x^(j k)/(j k (1 - x^(j k))), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d DivisorSigma[-1, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 21}]
nmax = 21; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[1, k]/k, j]*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; Take[CoefficientList[s, x], nmax + 1] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 03 2018 *)

E.g.f.: Product_{k>=1} (1 + x^k)^(A017665(k)/A017666(k)).
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(j*k*(1 - x^(j*k)))).
log(a(n)/n!) ~ sqrt(n/2) * Pi^2 / 3. - Vaclav Kotesovec, Sep 04 2018
a(n)/n! ~ c * exp(sqrt(n/2)*Pi^2/3) / n^(3/4 + log(2)/4), where c = 0.15653645678497413538057076667218805302154965061194080137... - Vaclav Kotesovec, Sep 05 2018

A338814 Expansion of e.g.f. log(Product_{k>0} (1 + x^k)^(1/k)).

Original entry on oeis.org

1, 0, 4, -6, 48, 0, 1440, -10080, 120960, 0, 7257600, -79833600, 958003200, 0, 348713164800, -3923023104000, 41845579776000, 0, 12804747411456000, -243290200817664000, 9731608032706560000, 0, 2248001455555215360000, -103408066955539906560000

Offset: 1

Seiichi Manyama, Nov 10 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 1..450

Column 1 of A338813.

Cf. A048272, A168243, A318249.

a[n_] := (n - 1)! * DivisorSum[n, (-1)^(# + 1) &]; Array[a, 25] (* Amiram Eldar, Apr 28 2021 *)

N=40; x='x+O('x^N); Vec(serlaplace(log(prod(k=1, N, (1+x^k)^(1/k)))))

{a(n) = if(n<1, 0, (n-1)!*sumdiv(n, d, (-1)^(d+1)))}

a(n) = (n-1)! * A048272(n).

A295794 Expansion of e.g.f. Product_{k>=1} exp(x^k/(1 + x^k)).

Original entry on oeis.org

1, 1, 1, 13, 25, 241, 2761, 14701, 153553, 1903105, 27877681, 263555821, 4788201001, 65083782193, 1040877257785, 24098794612621, 373918687272481, 7393663746307201, 164894196647876833, 3504497611085823565, 81863829346282866361, 2257321249626793901041, 49755091945025205954601

Offset: 0

Ilya Gutkovskiy, Nov 27 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..440

Cf. A028342, A048272, A168243, A206303, A294363, A294392, A295739.

a:=series(mul(exp(x^k/(1+x^k)),k=1..100),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 27 2019

nmax = 22; CoefficientList[Series[Product[Exp[x^k/(1 + x^k)], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[x D[Log[Product[(1 + x^k)^(1/k), {k, 1, nmax}]], x]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[-k Sum[(-1)^d, {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 22}]

E.g.f.: exp(Sum_{k>=1} A048272(k)*x^k).
E.g.f.: exp(x*f'(x)), where f(x) = log(Product_{k>=1} (1 + x^k)^(1/k)).
a(n) ~ exp(2*sqrt(n*log(2)) - 1/4 - n) * n^(n - 1/4) * log(2)^(1/4) / sqrt(2). - Vaclav Kotesovec, Sep 07 2018

cp's OEIS Frontend

A028342 Expansion of Product_{i>=1} (1 - x^i)^(-1/i); also of exp(Sum_{n>=1} (d(n)*x^n/n)) where d is number of divisors function.

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A206303 Expansion of e.g.f.: Product_{n>=1} (1 - x^(2*n-1))^(-1/(2*n-1)).

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A298906 Expansion of e.g.f. Product_{k>=1} (1 + x^k)^(1/k!).

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A356392 Expansion of e.g.f. ( Product_{k>0} (1+x^k)^(1/k) )^(1/(1-x)).

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A294356 E.g.f.: Product_{k>0} (1+x^k)^(-1/k).

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A295792 Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(1/k).

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A318696 Expansion of e.g.f. Product_{i>=1, j>=1} (1 + x^(i*j))^(1/(i*j)).

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A318769 Expansion of e.g.f. Product_{k>=1} (1 + x^k)^(sigma(k)/k), where sigma(k) is the sum of the divisors of k.

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A206303 Expansion of e.g.f.: Product_{n>=1} (1 - x^(2n-1))^(-1/(2n-1)).

A318696 Expansion of e.g.f. Product_{i>=1, j>=1} (1 + x^(ij))^(1/(ij)).