A269791 -id:A269791 - cp's OEIS frontend

A007841 Number of factorizations of permutations of n letters into cycles in nondecreasing length order.

1, 1, 3, 11, 56, 324, 2324, 18332, 167544, 1674264, 18615432, 223686792, 2937715296, 41233157952, 623159583552, 10008728738304, 171213653641344, 3092653420877952, 59086024678203264, 1185657912197967744, 25015435198774723584, 552130504313534175744

Offset: 0

Arnold Knopfmacher

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450
Vaclav Kotesovec, Graph - The asymptotic ratio
A. Knopfmacher, J. N. Ridley, Reciprocal sums over partitions and compositions, SIAM J. Discrete Math. 6 (1993), no. 3, 388-399.
D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388.

Cf. A007837, A007838, A249078, A249480, A249588, A249593, A269791, A269793, A269794.

p := product(1/(1-x^m/m), m=1..100):
s := series(p,x,100):
for i from 0 to 100 do printf(`%.0f,`,i!*coeff(s,x,i)) od:
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
     (i-1)!^j*b(n-i*j, i-1)*multinomial(n, n-i*j, i$j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 21 2014

nmax = 25; CoefficientList[Series[1/Product[(1 - x^k/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 24 2019 *)
nmax = 25; CoefficientList[Series[Exp[Sum[PolyLog[j, x^j]/j, {j, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 24 2019 *)

R(n,m):=if n=0 then 1 else if nVladimir Kruchinin, Sep 09 2014 */

N=66; q='q+O('q^N);
f=1/prod(n=1,N, 1-1/n*q^n );
egf=serlaplace(f);
Vec(egf)
/* Joerg Arndt, Oct 06 2012 */

E.g.f.: prod{m >= 1} 1/(1-x^m/m).
a(n) = Sum_{k=1..n} (n-1)!/(n-k)!*b(k)*a(n-k), where b(k) = Sum_{d divides k} d^(1-k/d) and a(0) = 1. - Vladeta Jovovic, Oct 14 2002
a(n) = R(n,1), R(n,m) = R(n,m+1)+binomial(n,m)*(m-1)!*R(n-m,m), R(n,n)=(n-1)!, R(n,m)=0 for nVladimir Kruchinin, Sep 09 2014
a(n) ~ c * n! * n, where c = exp(-gamma) = 0.56145948..., where gamma is the Euler-Mascheroni constant A001620 [Lehmer, 1972]. - Vaclav Kotesovec, Mar 05 2016
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*j^k)). - Ilya Gutkovskiy, May 27 2018

More terms from James Sellers, Jan 09 2001

Prepended a(0) = 1, Joerg Arndt, Oct 06 2012

A249588 G.f.: Product_{n>=1} 1/(1 - x^n/n^2) = Sum_{n>=0} a(n)*x^n/n!^2.

Original entry on oeis.org

1, 1, 5, 49, 856, 22376, 842536, 42409480, 2782192064, 229357803456, 23289083584704, 2851295406197184, 414855423241758720, 70695451937596732416, 13958230719814052097024, 3159974451734082088897536, 813380358295803762813321216, 236172126115504055456155975680

Offset: 0

Paul D. Hanna, Nov 01 2014

nonn

			G.f.: A(x) = 1 + x + 5*x^2/2!^2 + 49*x^3/3!^2 + 856*x^4/4!^2 +...
where
A(x) = 1/((1-x)*(1-x^2/4)*(1-x^3/9)*(1-x^4/16)*(1-x^5/25)*...).

Paul D. Hanna, Table of n, a(n) for n = 0..100
MathOverflow, asymptotic growth of a sum involving partitions, Sep 26 2021.

Cf. A249590, A249607, A007841, A249593, A249592, A269791, A269793, A269794.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+b(n-i, min(i, n-i))*((i-1)!*binomial(n, i))^2 ))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..17);  # Alois P. Heinz, Jul 27 2023

b[k_] := b[k] = DivisorSum[k, #^(1-2*k/#) &]; a[0] = 1; a[n_] := a[n] = Sum[n!*(n-1)!/(n-k)!^2*b[k]*a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 23 2015, adapted from PARI *)
Table[n!^2 * SeriesCoefficient[Product[1/(1 - x^m/m^2), {m, 1, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 05 2016 *)

{a(n)=n!^2*polcoeff(prod(k=1, n, 1/(1-x^k/k^2 +x*O(x^n))),n)}
for(n=0,20,print1(a(n),", "))

/* Using logarithmic derivative: */
{b(k) = sumdiv(k,d, d^(1-2*k/d))}
{a(n) = if(n==0,1,sum(k=1,n, n!*(n-1)!/(n-k)!^2 * b(k) * a(n-k)))}
for(n=0,20,print1(a(n),", "))

a(n) = Sum_{k=1..n} n!*(n-1)!/(n-k)!^2 * b(k) * a(n-k), where b(k) = Sum_{d|k} d^(1-2*k/d) and a(0) = 1 (after Vladeta Jovovic in A007841).

a(n) ~ 2 * n!^2. - Vaclav Kotesovec, Mar 05 2016

Name clarified by Vaclav Kotesovec, Mar 05 2016

A249593 G.f.: Product_{n>=1} 1/(1 - x^n/n^3) = Sum_{n>=0} a(n)*x^n/n!^3.

Original entry on oeis.org

1, 1, 9, 251, 16496, 2083824, 453803984, 156304214576, 80272385155584, 58631012094472704, 58713787327403063808, 78225670182020153384448, 135277046518915274471718912, 297374407080303931562525442048, 816367902369725640298981464096768

Offset: 0

Paul D. Hanna, Nov 02 2014

nonn

			G.f.: A(x) = 1 + x + 9*x^2/2!^3 + 251*x^3/3!^3 + 16496*x^4/4!^3 +...
where
A(x) = 1/((1-x)*(1-x^2/2^3)*(1-x^3/3^3)*(1-x^4/4^3)*(1-x^5/5^3)*...).

Vaclav Kotesovec, Table of n, a(n) for n = 0..180

Cf. A007841, A249588, A269791, A269793, A269794.

Table[n!^3 * SeriesCoefficient[Product[1/(1 - x^m/m^3), {m, 1, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 05 2016 *)

{a(n)=n!^3*polcoeff(prod(k=1, n, 1/(1-x^k/k^3 +x*O(x^n))),n)}
for(n=0,20,print1(a(n),", "))

/* Using logarithmic derivative: */
{b(k) = sumdiv(k, d, d^(1-3*k/d))}
{a(n) = if(n==0, 1, sum(k=1, n, n!^2*(n-1)!/(n-k)!^3 * b(k) * a(n-k)))}
for(n=0, 20, print1(a(n), ", "))

a(n) = Sum_{k=1..n} n!^2*(n-1)!/(n-k)!^3 * b(k) * a(n-k), where b(k) = Sum_{d|k} d^(1-3*k/d) and a(0) = 1 (after Vladeta Jovovic in A007841).

a(n) ~ c * n!^3, where c = Product_{k>=2} 1/(1-1/k^3) = 3*Pi/cosh(sqrt(3)*Pi/2) = 1.235488267746513477155075624616837... . - Vaclav Kotesovec, Mar 05 2016

Name clarified by Vaclav Kotesovec, Mar 05 2016

A269793 G.f.: Product_{n>=1} 1/(1 - x^n/n^5) = Sum_{n>=0} a(n)*x^n/n!^5.

Original entry on oeis.org

1, 1, 33, 8051, 8259776, 25822962624, 200839327164224, 3375758721819353792, 110621043661751405543424, 6532189550762931700406452224, 653226327065916563182761815212032, 105203470361723800472334968046839365632, 26178104032796403698593899646317901702496256

Offset: 0

Vaclav Kotesovec, Mar 05 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..120

Cf. A007841, A249588, A249593, A269791, A269794.

Table[n!^5 * SeriesCoefficient[Product[1/(1-x^k/k^5), {k, 1, n}], {x, 0, n}], {n, 0, 20}]

{a(n)=n!^5*polcoeff(prod(k=1, n, 1/(1-x^k/k^5 +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

a(n) ~ c * n!^5, where c = Product_{k>=2} 1/(1-1/k^5) = abs(Gamma((9+sqrt(5) + i*sqrt(10-2*sqrt(5)))/4) * Gamma((9-sqrt(5) + i*sqrt(10+2*sqrt(5)))/4))^2 = 1.03814501733099931382497266723652151296563..., where Gamma is the Gamma function and i is the imaginary unit. - Vaclav Kotesovec, Mar 05 2016

A269794 G.f.: Product_{n>=1} 1/(1 - x^n/n^6) = Sum_{n>=0} a(n)*x^n/n!^6.

Original entry on oeis.org

1, 1, 65, 47449, 194444416, 3038449102976, 141766192358448256, 16678817447073033946240, 4372271021740050216976646144, 2323608852183697867526563204694016, 2323611343146528421975097303187359268864, 4116421685969107286571222251382158945547976704

Offset: 0

Vaclav Kotesovec, Mar 05 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..100

Cf. A007841, A249588, A249593, A269791, A269793.

Table[n!^6 * SeriesCoefficient[Product[1/(1-x^k/k^6), {k, 1, n}], {x, 0, n}], {n, 0, 20}]

{a(n)=n!^6*polcoeff(prod(k=1, n, 1/(1-x^k/k^6 +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

a(n) ~ c * n!^6, where c = Product_{k>=2} 1/(1-1/k^6) = 6*Pi^2 / cosh(sqrt(3)*Pi/2)^2 = 1.0176208398261870492814795459985... . - Vaclav Kotesovec, Mar 05 2016

A336295 a(n) = (n!)^n * [x^n] Product_{k>=1} 1/(1 - x^k/k^n).

Original entry on oeis.org

1, 1, 5, 251, 359200, 25822962624, 141766192358448256, 83301485967496541735457536, 7013555995366382867427754604471779328, 109330254486209621988088555707809713786027354619904, 396335044092985772297627538614627390881554195217999599121962369024

Offset: 0

Ilya Gutkovskiy, Jul 16 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..30

Cf. A007841, A215910, A249588, A249593, A269791, A269793, A269794.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k)+b(n-i, min(n-i, i), k)*((i-1)!*binomial(n, i))^k))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..12);  # Alois P. Heinz, Jul 27 2023

Table[(n!)^n SeriesCoefficient[Product[1/(1 - x^k/k^n), {k, 1, n}], {x, 0, n}], {n, 0, 10}]

cp's OEIS Frontend

A007841 Number of factorizations of permutations of n letters into cycles in nondecreasing length order.

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A249588 G.f.: Product_{n>=1} 1/(1 - x^n/n^2) = Sum_{n>=0} a(n)*x^n/n!^2.

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A249593 G.f.: Product_{n>=1} 1/(1 - x^n/n^3) = Sum_{n>=0} a(n)*x^n/n!^3.

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A269793 G.f.: Product_{n>=1} 1/(1 - x^n/n^5) = Sum_{n>=0} a(n)*x^n/n!^5.

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A269794 G.f.: Product_{n>=1} 1/(1 - x^n/n^6) = Sum_{n>=0} a(n)*x^n/n!^6.

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A336295 a(n) = (n!)^n * [x^n] Product_{k>=1} 1/(1 - x^k/k^n).

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