A007838 -id:A007838 - cp's OEIS frontend

A305987 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k/k).

1, 1, 2, 9, 52, 355, 2976, 29897, 343988, 4423503, 63088600, 992691205, 17095554444, 319404545291, 6427307831504, 138546745515393, 3185841858310180, 77866726065935239, 2016161715005701128, 55127056896177521981, 1587073087715010466556, 47982707153606476112067, 1519931218769637781731712

Offset: 0

Ilya Gutkovskiy, Jun 15 2018

nonn

Stirling transform of A007838.

Vaclav Kotesovec, Table of n, a(n) for n = 0..420
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A007838, A305547, A305550, A305986.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(combinat[multinomial](n, n-i*j, i$j)/j!*
      b(n-i*j, i-1)*(i-1)!^j, j=0..min(1, n/i))))
    end:
a:= n-> add(Stirling2(n, j)*b(j$2), j=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 15 2018

nmax = 22; CoefficientList[Series[Product[(1 + (Exp[x] - 1)^k/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k + 1) (Exp[x] - 1)^(j k)/(k j^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
b[0] = 1; b[n_] := b[n] = Sum[(n - 1)!/(n - k)! DivisorSum[k, (-#)^(1 - k/#) &] b[n - k], {k, 1, n}]; a[n_] := a[n] = Sum[StirlingS2[n, k] b[k], {k, 0, n}]; Table[a[n], {n, 0, 22}]

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*(exp(x) - 1)^(j*k)/(k*j^k)).
a(n) = Sum_{k=0..n} Stirling2(n,k)*A007838(k).
a(n) ~ exp(-gamma) * n! / (2 * log(2)^(n+1)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 23 2019

A362362 Number of permutations of [n] such that each cycle contains its length as an element.

Original entry on oeis.org

1, 1, 1, 3, 8, 36, 174, 1104, 7440, 62640, 545040, 5649840, 60681600, 748621440, 9518342400, 136758585600, 2009451628800, 32848492723200, 549241915622400, 10066913176320000, 188293339922688000, 3832031198451456000, 79291640831090688000, 1771146970953744384000

Offset: 0

Alois P. Heinz, Jul 05 2023

nonn

The cycle lengths are distinct as a consequence of the definition.

			a(3) = 3: (123), (132), (1)(23).
a(4) = 8: (1234), (1243), (1324), (1342), (1423), (1432), (1)(234), (1)(243).

Alois P. Heinz, Table of n, a(n) for n = 0..450
Wikipedia, Permutation

Cf. A000009, A007838, A032020, A179973, A321520, A326493, A364277, A364406.

a:= n-> add((n-nops(p))!, p=select(l-> nops(l)=
        nops({l[]}), combinat[partition](n))):
seq(a(n), n=0..24);
# second Maple program:
b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$3):
seq(a(n), n=0..24);

b[n_, i_, p_] := b[n, i, p] = If[i*(i + 1)/2 < n, 0, If[n == 0, p!, b[n, i - 1, p] + b[n - i, Min[n - i, i - 1], p - 1]]];
a[n_] := b[n, n, n];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Nov 15 2023, from second Maple program *)

A087639 E.g.f.: Product_{m >= 1} (1+x^(2m)/(2m)) (even powers only).

Original entry on oeis.org

1, 1, 6, 210, 8400, 740880, 88814880, 15217282080, 3319002086400, 992431440000000, 351841557779712000, 156995673442223616000, 82429416503416958976000, 52017974139195896832000000, 37547796668359538444083200000, 31987697744989345038846566400000

Offset: 0

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 06 2003

nonn

Number of permutations of 2*n elements with distinct cycle lengths and without odd cycles. - Vladeta Jovovic, Aug 17 2004

Alois P. Heinz, Table of n, a(n) for n = 0..225

Cf. A007838, A007841, A088994, A294506, A305199, A309319.

b:= proc(n, i) option remember; `if`((i/2)*(i/2+1)n, 0, (i-1)!*
       b(n-i, i-2)*binomial(n, i))))
    end:
a:= n-> b(2*n$2):
seq(a(n), n=0..17);  # Alois P. Heinz, Nov 01 2017

nmax = 20; Table[(CoefficientList[Series[Product[1 + x^(2*k)/(2*k), {k, 1, 2*nmax}], {x, 0, 2*nmax}], x]*Range[0, 2*nmax]!)[[2*n + 1]], {n, 0, nmax}] (* Vaclav Kotesovec, Jul 23 2019 *)

a(n) ~ 2*exp(-gamma/2) * (2*n)! / (Pi*sqrt(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 23 2019

More terms from Christian G. Bower, Jan 06 2006

A292358 E.g.f.: Product_{k>=1} 1/(1 + x^k/k).

Original entry on oeis.org

1, -1, 1, -5, 20, -104, 584, -4304, 35720, -329160, 3239112, -36135912, 438454752, -5743527360, 80351263680, -1218698312064, 19599583392384, -334335747652224, 6019295318075520, -114911886106373760, 2305234779285164544, -48596575400453366784

Offset: 0

Seiichi Manyama, Sep 14 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A007838, A007841.

nmax = 20; CoefficientList[Series[Product[1/(1 + x^k/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 15 2017 *)

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

a(n) ~ (-1)^n * n!. - Vaclav Kotesovec, Sep 15 2017

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^k*x^(j*k)/(k*j^k)). - Ilya Gutkovskiy, May 27 2018

A292359 E.g.f.: Product_{k>=1} (1 - x^k/k).

Original entry on oeis.org

1, -1, -1, 1, 2, 26, -6, 414, -624, -288, -20880, 164880, -9756000, 43529760, -324404640, -4052492640, -48521410560, 1168445053440, -26858914467840, 341240066334720, -5752671815116800, 49267037136844800, -769468911734476800, 39863275492626432000

Offset: 0

Seiichi Manyama, Sep 14 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..451

Cf. A007838, A007841.

nmax = 20; CoefficientList[Series[Product[(1 - x^k/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 15 2017 *)

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

Conjecture: a(n) ~ -(-1)^n * n! / n^2. - Vaclav Kotesovec, Sep 15 2017

E.g.f.: exp(-Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*j^k)). - Ilya Gutkovskiy, May 27 2018

A326864 G.f.: Product_{k>=1} (1 + x^k/k^2) = Sum_{n>=0} a(n)*x^n/n!^2.

Original entry on oeis.org

1, 1, 1, 13, 100, 1876, 57636, 2051316, 104640768, 6819033600, 576652089600, 57187381536000, 7057192160793600, 1014733052692300800, 172646881540527744000, 33848454886497227289600, 7637231669166956976537600, 1948418678155880277481881600

Offset: 0

Vaclav Kotesovec, Jul 27 2019

nonn

			a(n) ~ c * (n-1)!^2, where c = A156648 = Product_{k>=1} (1 + 1/k^2) = sinh(Pi)/Pi = 3.67607791037497772...

Alois P. Heinz, Table of n, a(n) for n = 0..254

Cf. A007838, A249588, A326865.

Cf. A087132.

b:= proc(n, i) option remember; `if`(i*(i+1)/2 b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 27 2023

nmax = 20; CoefficientList[Series[Product[(1+x^k/k^2), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!^2

A364279 Number of permutations of [n] with distinct cycle lengths such that no cycle contains its length as an element.

Original entry on oeis.org

1, 0, 0, 1, 2, 12, 86, 546, 4284, 39588, 416988, 4378848, 54297504, 695592000, 9840307680, 149031686880, 2387863575360, 40338090711360, 736126007279040, 13938942123429120, 279358800902737920, 5894877845100625920, 129943826126987765760, 2985640822908446976000

Offset: 0

Alois P. Heinz, Jul 17 2023

nonn

			a(3) = 1: (13)(2).
a(4) = 2: (124)(3), (142)(3).
a(5) = 12: (1235)(4), (1253)(4), (1325)(4), (1352)(4), (1523)(4), (1532)(4), (124)(35), (142)(35), (125)(34), (152)(34), (13)(245), (13)(254).

Wikipedia, Permutation

Cf. A000009, A007838, A362362, A364277, A364278, A364281, A364406.

A326857 E.g.f.: Product_{k>=1} (1 + x^(3k-1) / (3k-1)).

Original entry on oeis.org

1, 0, 1, 0, 0, 24, 0, 504, 5040, 0, 226800, 3628800, 0, 438721920, 6227020800, 16345929600, 1127656857600, 20922789888000, 58203397324800, 6697914906009600, 121645100408832000, 655224745383936000, 51359276952023040000, 1124000727777607680000

Offset: 0

Vaclav Kotesovec, Jul 27 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..440
Vaclav Kotesovec, Graph - the asymptotic ratio (30000 terms)

Cf. A007838, A088994, A326755, A326858.

nmax = 30; CoefficientList[Series[Product[(1+x^(3*k-1)/(3*k-1)), {k, 1, Floor[nmax/3]+1}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) ~ 2 * Pi * n! / (exp(gamma/3) * 3^(5/6) * Gamma(1/3)^2 * n^(2/3)), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function.

A335638 Expansion of e.g.f. Product_{k>0} (1 + tan(x)^k / k).

Original entry on oeis.org

1, 1, 1, 7, 22, 190, 1170, 11646, 109520, 1289168, 16018064, 223757840, 3407971488, 55709905056, 998011344928, 18778681069024, 385316251841536, 8225863823985664, 189755182485906432, 4538893733746003968, 116147781156885837824, 3078530007519830730752, 86521073899573883088896

Offset: 0

Seiichi Manyama, Oct 03 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..430

Cf. A000182, A007838, A335630, A335636, A335637, A336046.

max = 22; Range[0, max]! * CoefficientList[Series[Product[1 + Tan[x]^k/k, {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Oct 03 2020 *)

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

N=40; x='x+O('x^N); Vec(serlaplace(exp(sum(i=1, N, sum(j=1, N\i, (-1)^(i+1)*tan(x)^(i*j)/(i*j^i))))))

E.g.f.: exp( Sum_{i>0} Sum_{j>0} (-1)^(i+1)*tan(x)^(i*j)/(i*j^i) ).

Conjecture: a(n) ~ A080130 * 2^(2*n+1) * n! / Pi^(n+1). - Vaclav Kotesovec, Oct 04 2020

A182965 E.g.f.: A(x) = Product_{n>=1} (1 + 2*x^n/n)^n.

Original entry on oeis.org

1, 2, 4, 36, 168, 1440, 13920, 134400, 1619520, 20549760, 294631680, 4449096960, 74429752320, 1312794362880, 24870628823040, 501316411115520, 10661299747338240, 239672059847700480, 5664762159214878720

Offset: 0

Paul D. Hanna, Dec 19 2010

nonn

			E.g.f.: A(x) = 1 + 2*x + 4*x^2/2! + 36*x^3/3! + 168*x^4/4! +...
A(x) = (1+2x)*(1+2x^2/2)^2*(1+2x^3/3)^3*(1+2x^4/4)^4*(1+2x^5/5)^5*...

Vaclav Kotesovec, Table of n, a(n) for n = 0..439

Cf. A181541, A182966, A182967, A007838.

nmax = 20; CoefficientList[Series[Product[(1 + 2*x^k/k)^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 07 2020 *)

{a(n,k=2)=n!*polcoeff(prod(m=1,n,(1+k*x^m/m+x*O(x^n))^m),n)}

cp's OEIS Frontend

A305987 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k/k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A362362 Number of permutations of [n] such that each cycle contains its length as an element.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A087639 E.g.f.: Product_{m >= 1} (1+x^(2*m)/(2*m)) (even powers only).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A292358 E.g.f.: Product_{k>=1} 1/(1 + x^k/k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A292359 E.g.f.: Product_{k>=1} (1 - x^k/k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A326864 G.f.: Product_{k>=1} (1 + x^k/k^2) = Sum_{n>=0} a(n)*x^n/n!^2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A364279 Number of permutations of [n] with distinct cycle lengths such that no cycle contains its length as an element.

Views

Author

Keywords

Examples

Links

Crossrefs

A326857 E.g.f.: Product_{k>=1} (1 + x^(3*k-1) / (3*k-1)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A087639 E.g.f.: Product_{m >= 1} (1+x^(2m)/(2m)) (even powers only).

A326857 E.g.f.: Product_{k>=1} (1 + x^(3k-1) / (3k-1)).