cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 44 results. Next

A088994 Number of permutations in the symmetric group S_n such that the size of their centralizer is odd.

Original entry on oeis.org

1, 1, 0, 2, 8, 24, 144, 720, 8448, 64512, 576000, 5529600, 74972160, 887546880, 11285084160, 168318259200, 2843121254400, 44790578380800, 747955947110400, 13937735643955200, 287117441217331200, 5838778006909747200, 120976472421826560000, 2712639152754878054400
Offset: 0

Views

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 01 2003

Keywords

Comments

a(n) is the number of n-permutations composed only of odd cycles of distinct length. - Geoffrey Critzer, Mar 08 2013
Also the number of permutations p of [n] with unique (functional) square root, i.e., there exists a unique permutation g such that g^2 = p. - Keith J. Bauer, Jan 08 2024

Links

Crossrefs

Cf. A000142, A007838, A007841, A087639, A088335, A294506, A305199, A309319.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(((i+1)/2)^2n, 0, (i-1)!*
       b(n-i, i-2)*binomial(n, i))))
    end:
a:= n-> b(n, n-1+irem(n, 2)):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 01 2017
  • Mathematica
    nn=20;Range[0,nn]!CoefficientList[Series[Product[1+x^(2i-1)/(2i-1),{i,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Mar 08 2013 *)
  • PARI
    {a(n)=n!*polcoeff( prod(k=1, n, 1+(k%2)*x^k/k, 1+x*O(x^n)), n)} /* Michael Somos, Sep 19 2006 */

Formula

E.g.f.: Product_{m >= 1} (1+x^(2*m-1)/(2*m-1)). - Vladeta Jovovic, Nov 05 2003
a(n) ~ exp(-gamma/2) * n! / sqrt(2*n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 23 2019
a(n) = n! - A088335(n). - Alois P. Heinz, Jan 27 2020

Extensions

More terms from Vladeta Jovovic, Nov 03 2003
a(0)=1 prepended by Seiichi Manyama, Nov 01 2017

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

Views

Author

Paul D. Hanna, Nov 01 2014

Keywords

Examples

			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)*...).

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    {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),", "))
  • PARI
    /* 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),", "))

Formula

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

Extensions

Name clarified by Vaclav Kotesovec, Mar 05 2016

A249078 E.g.f.: exp(1)*P(x) - Q(x), where P(x) = 1/Product_{n>=1} (1 - x^n/n) and Q(x) = Sum_{n>=1} 1/Product_{k=1..n} (k - x^k).

Original entry on oeis.org

1, 1, 4, 17, 96, 595, 4516, 37104, 351020, 3604001, 41007240, 502039444, 6703536516, 95376507135, 1459072099824, 23677731306350, 408821193129564, 7443839953433701, 143258713990271960, 2893053522512463984, 61396438056305204020, 1362146168353191078195, 31605702195327725326560
Offset: 0

Views

Author

Paul D. Hanna, Oct 28 2014

Keywords

Comments

The function P(x) = Product_{n>=1} 1/(1 - x^n/n) equals the e.g.f. of A007841, the number of factorizations of permutations of n letters into cycles in nondecreasing length order.

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 17*x^3/3! + 96*x^4/4! + 595*x^5/5! +...
such that A(x) = exp(1)*P(x) - Q(x), where
P(x) = 1/Product_{n>=1} (1 - x^n/n) = Sum_{n>=0} A007841(n)*x^n/n!, and
Q(x) = Sum_{n>=1} 1/Product_{k=1..n} (k - x^k).
More explicitly,
P(x) = 1/((1-x)*(1-x^2/2)*(1-x^3/3)*(1-x^4/4)*(1-x^5/5)*...);
Q(x) = 1/(1-x) + 1/((1-x)*(2-x^2)) + 1/((1-x)*(2-x^2)*(3-x^3)) + 1/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)) + 1/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)*(5-x^5)) +...
We can illustrate the initial terms a(n) in the following manner.
The coefficients in Q(x) = Sum_{n>=0} q(n)*x^n/n! begin:
q(0) = 1.7182818284590452...
q(1) = 1.7182818284590452...
q(2) = 4.1548454853771357...
q(3) = 12.901100113049497...
q(4) = 56.223782393706533...
q(5) = 285.72331242073065...
q(6) = 1801.2869693388211...
q(7) = 12727.542479311217...
q(8) = 104411.81066734227...
q(9) = 947120.40724315491...
and the coefficients in P(x) = 1/Product_{n>=1} (1 - x^n/n) begin:
A007841 = [1, 1, 3, 11, 56, 324, 2324, 18332, 167544, ...];
from which we can generate this sequence like so:
a(0) = exp(1)*1 - q(0) = 1;
a(1) = exp(1)*1 - q(1) = 1;
a(2) = exp(1)*3 - q(2) = 4;
a(3) = exp(1)*11 - q(3) = 17;
a(4) = exp(1)*56 - q(4) = 96;
a(5) = exp(1)*324 - q(5) = 595;
a(6) = exp(1)*2324 - q(6) = 4516;
a(7) = exp(1)*18332 - q(7) = 37104;
a(8) = exp(1)*167544 - q(8) = 351020; ...

Links

Crossrefs

Cf. A007841, A249474, A249475, A249476, A249477, A249478, A249480.

Programs

  • PARI
    \p100 \\ set precision
{P=Vec(serlaplace(prod(k=1,31,1/(1-x^k/k +O(x^31)))));} \\ A007841
{Q=Vec(serlaplace(sum(n=1,201,prod(k=1,n,1./(k-x^k +O(x^31))))));}
for(n=0,30,print1(round(exp(1)*P[n+1]-Q[n+1]),", "))

A305199 Expansion of e.g.f. Product_{k>=1} (1 + x^k/k)/(1 - x^k/k).

Original entry on oeis.org

1, 2, 6, 28, 152, 1008, 7756, 67688, 659424, 7123776, 84154224, 1079913888, 14962632384, 222447507072, 3531920599008, 59664827178048, 1067975819206656, 20192760528611328, 402169396496004864, 8414121277765679616, 184498963978904644608, 4231186653661629843456
Offset: 0

Views

Author

Ilya Gutkovskiy, May 27 2018

Keywords

Comments

Exponential convolution of the sequences A007838 and A007841.

Links

Crossrefs

Cf. A007838, A007841, A292358, A292359, A295792.

Programs

  • Maple
    a:=series(mul((1+x^k/k)/(1-x^k/k),k=1..100),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 21; CoefficientList[Series[Product[(1 + x^k/k)/(1 - x^k/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[Sum[(1 + (-1)^(k + 1)) x^(j k)/(k j^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (1 + (-1)^(k+1))*x^(j*k)/(k*j^k)).
a(n) ~ sqrt(Pi/2) * n^(n + 5/2) / exp(n + 2*gamma), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Mar 26 2019

A294506 E.g.f.: 1/Product_{k>0} (1-x^(2*k-1)/(2*k-1)).

Original entry on oeis.org

1, 1, 2, 8, 32, 184, 1184, 9008, 74752, 726528, 7583232, 87931392, 1092516864, 14863589376, 215094226944, 3358032635904, 55181218873344, 970561417248768, 17945595514847232, 351221170194874368, 7186120683011702784, 155103171658691641344
Offset: 0

Views

Author

Seiichi Manyama, Nov 01 2017

Keywords

Links

Crossrefs

Cf. A007838, A007841, A087639, A088994, A309319.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[1/Product[(1-x^(2*k-1)/(2*k-1)), {k, 1, Floor[nmax/2] + 1}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 02 2017 *)

Formula

a(n) ~ 2*exp(-gamma/2) * sqrt(2*n) * n! / Pi, where gamma is the Euler-Mascheroni constant A001620 [Lehmer, 1972]. - Vaclav Kotesovec, Jul 23 2019

A249474 E.g.f.: P(x)/exp(1) + Q(x), where P(x) = 1/Product_{n>=1} (1 - x^n/n) and Q(x) = Sum_{n>=1} -(-1)^n / Product_{k=1..n} (k - x^k).

Original entry on oeis.org

1, 1, 2, 7, 30, 169, 1128, 8700, 76494, 753139, 8182188, 97131376, 1256860330, 17470791933, 261284377168, 4164406202270, 70677340199670, 1268718107324255, 24091289738163140, 480954355282406340, 10097484764045220626, 221918808641500960217, 5103937368681669463800
Offset: 0

Views

Author

Paul D. Hanna, Oct 29 2014

Keywords

Comments

The function P(x) = Product_{n>=1} 1/(1 - x^n/n) equals the e.g.f. of A007841, the number of factorizations of permutations of n letters into cycles in nondecreasing length order.

Examples

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 30*x^4/4! + 169*x^5/5! +...
such that A(x) = exp(-1)*P(x) + Q(x), where
P(x) = 1/Product_{n>=1} (1 - x^n/n) = Sum_{n>=0} A007841(n)*x^n/n!, and
Q(x) = Sum_{n>=1} -(-1)^n / Product_{k=1..n} (k - x^k).
More explicitly,
P(x) = 1/((1-x)*(1-x^2/2)*(1-x^3/3)*(1-x^4/4)*(1-x^5/5)*...);
Q(x) = 1/(1-x) - 1/((1-x)*(2-x^2)) + 1/((1-x)*(2-x^2)*(3-x^3)) - 1/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)) + 1/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)*(5-x^5)) +-...
We can illustrate the initial terms a(n) in the following manner.
The coefficients in Q(x) = Sum_{n>=0} q(n)*x^n/n! begin:
q(0) = 0.632120558828557678...
q(1) = 0.632120558828557678...
q(2) = 0.896361676485673035...
q(3) = 2.953326147114134462...
q(4) = 9.398751294399229990...
q(5) = 49.80706106045268780...
q(6) = 273.0481787175680446...
q(7) = 1956.034084445119360...
q(8) = 14858.00690837186767...
q(9) = 137211.6953065362928...
and the coefficients in P(x) = 1/Product_{n>=1} (1 - x^n/n) begin:
A007841 = [1, 1, 3, 11, 56, 324, 2324, 18332, 167544, ...];
from which we can generate this sequence like so:
a(0) = exp(-1)*1 + q(0) = 1;
a(1) = exp(-1)*1 + q(1) = 1;
a(2) = exp(-1)*3 + q(2) = 2;
a(3) = exp(-1)*11 + q(3) = 7;
a(4) = exp(-1)*56 + q(4) = 30;
a(5) = exp(-1)*324 + q(5) = 169;
a(6) = exp(-1)*2324 + q(6) = 1128;
a(7) = exp(-1)*18332 + q(7) = 8700;
a(8) = exp(-1)*167544 + q(8) = 76494; ...

Links

Crossrefs

Cf. A007841, A249078, A249475, A249476, A249477, A249478, A249480.

Programs

  • PARI
    \p100 \\ set precision
{P=Vec(serlaplace(prod(k=1, 31, 1/(1-x^k/k +O(x^31))))); } \\ A007841
{Q=Vec(serlaplace(sum(n=1, 201, -(-1)^n * prod(k=1, n, 1./(k-x^k +O(x^31)))))); }
for(n=0, 30, print1(round(exp(-1)*P[n+1]+Q[n+1]), ", "))

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

Views

Author

Paul D. Hanna, Nov 02 2014

Keywords

Examples

			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)*...).

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    {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),", "))
  • PARI
    /* 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), ", "))

Formula

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

Extensions

Name clarified by Vaclav Kotesovec, Mar 05 2016

A294469 E.g.f.: 1/Product_{k>0} (1 - x^k/k)^k.

Original entry on oeis.org

1, 1, 4, 18, 114, 810, 7140, 68880, 766920, 9304680, 125086080, 1814015280, 28588356720, 481128888240, 8678237087520, 166041500264640, 3371031116893440, 72153115744469760, 1627441316510929920, 38500269726897538560, 954533425718494702080
Offset: 0

Views

Author

Seiichi Manyama, Oct 31 2017

Keywords

Links

Crossrefs

Cf. A007841, A181541, A294470, A294471.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[1/(1-x^k/k)^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 01 2017 *)
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, (1-x^k/k)^k)))

Formula

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*j^(k-1))). - Ilya Gutkovskiy, Sep 12 2018

A309319 E.g.f.: 1/Product_{k>0} (1 - x^(2*k)/(2*k)) (even powers only).

Original entry on oeis.org

1, 1, 12, 300, 15960, 1232280, 157006080, 25418352960, 5859886032000, 1655203620470400, 604893737678630400, 261278195494386470400, 140231830875916632652800, 86107922772424330377600000, 63316800257542340301112320000, 52666943508290765740968161280000
Offset: 0

Views

Author

Vaclav Kotesovec, Jul 23 2019

Keywords

Links

Crossrefs

Cf. A007838, A007841, A087639, A088994, A294506.

Programs

  • Mathematica
    nmax = 20; Table[(CoefficientList[Series[1/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}]

Formula

a(n) ~ exp(-gamma/2) * (2*n)! / sqrt(n) [Lehmer, 1972], where gamma is the Euler-Mascheroni constant A001620.

A249475 E.g.f.: exp(2)*P(x) - Q(x), where P(x) = 1/Product_{n>=1} (1 - x^n/n) and Q(x) = Sum_{n>=1} 2^n/Product_{k=1..n} (k - x^k).

Original entry on oeis.org

1, 1, 5, 25, 156, 1048, 8400, 72384, 710184, 7519240, 87797880, 1098513880, 14945280640, 216079283040, 3352657547680, 55071779464352, 961293645943680, 17669716422651776, 342988501737128576, 6978772157389361280, 149123855108936024576, 3328674238745847019520
Offset: 0

Views

Author

Paul D. Hanna, Oct 29 2014

Keywords

Comments

The function P(x) = Product_{n>=1} 1/(1 - x^n/n) equals the e.g.f. of A007841, the number of factorizations of permutations of n letters into cycles in nondecreasing length order.

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 25*x^3/3! + 156*x^4/4! + 1048*x^5/5! +...
such that A(x) = exp(2)*P(x) - Q(x), where
P(x) = 1/Product_{n>=1} (1 - x^n/n) = Sum_{n>=0} A007841(n)*x^n/n!, and
Q(x) = Sum_{n>=1} 2^n / Product_{k=1..n} (k - x^k).
More explicitly,
P(x) = 1/((1-x)*(1-x^2/2)*(1-x^3/3)*(1-x^4/4)*(1-x^5/5)*...);
Q(x) = 2/(1-x) + 2^2/((1-x)*(2-x^2)) + 2^3/((1-x)*(2-x^2)*(3-x^3)) + 2^4/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)) + 2^5/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)*(5-x^5)) +...
We can illustrate the initial terms a(n) in the following manner.
The coefficients in Q(x) = Sum_{n>=0} q(n)*x^n/n! begin:
q(0) = 6.3890560989306502272...
q(1) = 6.3890560989306502272...
q(2) = 17.167168296791950681...
q(3) = 56.279617088237152499...
q(4) = 257.78714154011641272...
q(5) = 1346.0541760535306736...
q(6) = 8772.1663739148311280...
q(7) = 63072.176405596679965...
q(8) = 527808.01503923686167...
q(9) = 4851990.6204200261720...
and the coefficients in P(x) = 1/Product_{n>=1} (1 - x^n/n) begin:
A007841 = [1, 1, 3, 11, 56, 324, 2324, 18332, 167544, ...];
from which we can generate this sequence like so:
a(0) = exp(2)*1 - q(0) = 1;
a(1) = exp(2)*1 - q(1) = 1;
a(2) = exp(2)*3 - q(2) = 5;
a(3) = exp(2)*11 - q(3) = 25;
a(4) = exp(2)*56 - q(4) = 156;
a(5) = exp(2)*324 - q(5) = 1048;
a(6) = exp(2)*2324 - q(6) = 8400;
a(7) = exp(2)*18332 - q(7) = 72384;
a(8) = exp(2)*167544 - q(8) = 710184; ...

Links

Crossrefs

Cf. A007841, A249078, A249474, A249476, A249477, A249478, A249480.

Programs

  • PARI
    \p100 \\ set precision
{P=Vec(serlaplace(prod(k=1, 31, 1/(1-x^k/k +O(x^31))))); } \\ A007841
{Q=Vec(serlaplace(sum(n=1, 201, 2^n * prod(k=1, n, 1./(k-x^k +O(x^31)))))); }
for(n=0, 30, print1(round(exp(2)*P[n+1]-Q[n+1]), ", "))
Showing 1-10 of 44 results. Next

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