A246609 -id:A246609 - cp's OEIS frontend

A060435 Number of functions f: {1,2,...,n} -> {1,2,...,n} with even cycles only.

1, 0, 1, 6, 57, 680, 9945, 172032, 3438673, 78003648, 1980083025, 55616359040, 1712630427849, 57375166877184, 2077563829893097, 80859304977696000, 3366275257190794785, 149270897223530835968, 7024011523121427204897, 349574012216588890718208

Offset: 0

Vladeta Jovovic, Apr 07 2001

nonn

E.g.f. equals the square-root of the e.g.f. of A134095. - Paul D. Hanna, Oct 11 2007

			E.g.f. A(x) = 1 + 0*x + 1*x^2/2! + 6*x^3/3! + 57*x^4/4! + 680*x^5/5! +...
The formula A(x) = 1/sqrt(1 - LambertW(-x)^2 ) is illustrated by:
A(x) = 1/sqrt(1 - (x+ x^2+ 3^2*x^3/3!+ 4^3*x^4/4!+ 5^4*x^5/5! +...)^2).

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

Alois P. Heinz, Table of n, a(n) for n = 0..386

Cf. A134095.

Column k=2 of A246609.

t = Sum[n^(n - 1) x^n/n!, {n, 1, 20}]; Range[0, 20]! CoefficientList[Series[(1/(1 - t^2))^(1/2), {x, 0, 20}], x]  (* Geoffrey Critzer, Dec 07 2011 *)

{a(n)=local(LambertW=sum(k=0,n,(-x)^(k+1)*(k+1)^k/(k+1)!) +x*O(x^n)); n!*polcoeff(1/sqrt(1-subst(LambertW,x,-x)^2),n)} \\ Paul D. Hanna, Oct 11 2007

E.g.f.: 1/sqrt(1-(LambertW(-x))^2). a(n)=(n-1)!*Sum_{k=0..floor((n-2)/2)} (k+1)/2^(2*k+1)*binomial(2*k+2, k+1)*n^(n-2-2*k)/(n-2-2*k)!.
A134095(n) = Sum_{k=0..n} C(n,k) * a(n-k) * a(k) with a(0)=1 and a(1)=0 where A134095(n) = Sum_{k=0..n} C(n,k) * (n-k)^k * k^(n-k). - Paul D. Hanna, Oct 11 2007
a(n) ~ n! * 2^(3/4)*Gamma(3/4)*exp(n)/(4*Pi*n^(3/4)) * (1- 5*Pi/ (24*Gamma(3/4)^2*sqrt(n))). - Vaclav Kotesovec, Sep 24 2013

More terms from Alois P. Heinz, Aug 26 2014

A246610 Number of endofunctions on [n] whose cycle lengths are multiples of 3.

Original entry on oeis.org

1, 0, 0, 2, 24, 300, 4480, 78750, 1591296, 36355256, 927244800, 26127386010, 806251494400, 27046291980708, 980094896062464, 38158333538165750, 1588601646620835840, 70427042234715548400, 3312574102411273437184, 164767312911755127462066, 8641342923227371929600000

Offset: 0

Alois P. Heinz, Aug 31 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300

Column k=3 of A246609.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, add(b(n-i*j, i+3)*(i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
    end:
a:= a->add(b(j, 3)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..20);

CoefficientList[Series[1/(1+LambertW[-x]^3)^(1/3),{x,0,20}],x] * Range[0,20]!  (* Vaclav Kotesovec, Sep 01 2014 *)

E.g.f.: 1/(1+LambertW(-x)^3)^(1/3). - Vaclav Kotesovec, Sep 01 2014

a(n) ~ Gamma(5/6) * (n^(n-1/3) / (12^(1/3) * sqrt(Pi))) * (1 - 2^(7/6) * Gamma(1/3)^3 / (9 * Pi^(3/2) * sqrt(n))). - Vaclav Kotesovec, Sep 01 2014

A246611 Number of endofunctions on [n] whose cycle lengths are multiples of 4.

Original entry on oeis.org

1, 0, 0, 0, 6, 120, 2160, 41160, 866460, 20294064, 526680000, 15036999120, 468848156040, 15859299473160, 578619457031616, 22654279249875000, 947570269816868880, 42174922731482980320, 1990416896317283627520, 99290011292792071612704, 5220362654145754082460000

Offset: 0

Alois P. Heinz, Aug 31 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300

Column k=4 of A246609.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, add(b(n-i*j, i+4)*(i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
    end:
a:= a->add(b(j, 4)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..20);

CoefficientList[Series[1/(1-LambertW[-x]^4)^(1/4),{x,0,20}],x] * Range[0,20]!  (* Vaclav Kotesovec, Sep 01 2014 *)

E.g.f.: 1/(1-LambertW(-x)^4)^(1/4). - Vaclav Kotesovec, Sep 01 2014

a(n) ~ n^(n-3/8) * (sqrt(Pi) / (2^(1/8) * Gamma(1/8))) * (1 - 11 * sqrt(2/n) * Gamma(1/8) / (64 * Gamma(5/8))). - Vaclav Kotesovec, Sep 01 2014

A246612 Number of endofunctions on [n] whose cycle lengths are multiples of 5.

Original entry on oeis.org

1, 0, 0, 0, 0, 24, 720, 17640, 430080, 11022480, 302835456, 8976567600, 287236730880, 9901354268520, 366504424366080, 14515574208519384, 612912858447052800, 27497080096591516320, 1306520704652165775360, 65556843230110158804960, 3464412427539012262035456

Offset: 0

Alois P. Heinz, Aug 31 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300

Column k=5 of A246609.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, add(b(n-i*j, i+5)*(i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
    end:
a:= a->add(b(j, 5)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);

CoefficientList[Series[1/(1+LambertW[-x]^5)^(1/5),{x,0,20}],x] * Range[0,20]!  (* Vaclav Kotesovec, Sep 01 2014 *)

E.g.f.: 1/(1+LambertW(-x)^5)^(1/5). - Vaclav Kotesovec, Sep 01 2014

a(n) ~ n^(n-2/5) * 2^(2/5) * (sqrt(Pi) / (5^(1/5) * Gamma(1/10))) * (1 - 14 * sqrt(2/n) * Gamma(1/10) / (75 * Gamma(3/5))). - Vaclav Kotesovec, Sep 01 2014

A246613 Number of endofunctions on [n] whose cycle lengths are multiples of 6.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 120, 5040, 161280, 4898880, 151200000, 4870182240, 165589522560, 5970883913280, 228632817853440, 9292843409850000, 400385970127503360, 18252778818295082880, 878619947953407528960, 44562094805330883023040, 2376296054962478899200000

Offset: 0

Alois P. Heinz, Aug 31 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300

Column k=6 of A246609.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, add(b(n-i*j, i+6)*(i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
    end:
a:= a->add(b(j, 6)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);

CoefficientList[Series[1/(1-LambertW[-x]^6)^(1/6),{x,0,20}],x] * Range[0,20]!  (* Vaclav Kotesovec, Sep 01 2014 *)

E.g.f.: 1/(1-LambertW(-x)^6)^(1/6). - Vaclav Kotesovec, Sep 01 2014

a(n) ~ n^(n-5/12) * 2^(1/4) * (sqrt(Pi) / (3^(1/6) * Gamma(1/12))) * (1 - 85 * sqrt(2/n) * Gamma(1/12) / (432 * Gamma(7/12))). - Vaclav Kotesovec, Sep 01 2014

A246614 Number of endofunctions on [n] whose cycle lengths are multiples of 7.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 720, 40320, 1632960, 60480000, 2213719200, 82771476480, 3211179491520, 130247609057280, 5542874078742000, 247828800444825600, 11643316737494682240, 574455837701776343040, 29734324379435446505280, 1612668074301849600000000

Offset: 0

Alois P. Heinz, Aug 31 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300

Column k=7 of A246609.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, add(b(n-i*j, i+7)*(i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
    end:
a:= a->add(b(j, 7)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);

CoefficientList[Series[1/(1+LambertW[-x]^7)^(1/7),{x,0,20}],x] * Range[0,20]!  (* Vaclav Kotesovec, Sep 01 2014 *)

E.g.f.: 1/(1+LambertW(-x)^7)^(1/7). - Vaclav Kotesovec, Sep 01 2014

a(n) ~ n^(n-3/7) * 2^(3/7) * (sqrt(Pi) / (7^(1/7) * Gamma(1/14))) * (1 - 10 * sqrt(2/n) * Gamma(1/14) / (49 * Gamma(4/7))). - Vaclav Kotesovec, Sep 01 2014

A246615 Number of endofunctions on [n] whose cycle lengths are multiples of 8.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 5040, 362880, 18144000, 804988800, 34488115200, 1482082842240, 65120246231040, 2955402450000000, 139297568464454400, 6837892496270035200, 350034595015623598080, 18692451288585606954240, 1041114148173066240000000

Offset: 0

Alois P. Heinz, Aug 31 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300

Column k=8 of A246609.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, add(b(n-i*j, i+8)*(i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
    end:
a:= a->add(b(j, 8)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);

CoefficientList[Series[1/(1-LambertW[-x]^8)^(1/8),{x,0,20}],x] * Range[0,20]!  (* Vaclav Kotesovec, Sep 01 2014 *)

E.g.f.: 1/(1-LambertW(-x)^8)^(1/8). - Vaclav Kotesovec, Sep 01 2014

a(n) ~ n^(n-7/16) * 2^(1/16) * (sqrt(Pi) / Gamma(1/16)) * (1 - 161 * sqrt(2/n) * Gamma(1/16) / (768 * Gamma(9/16))). - Vaclav Kotesovec, Sep 01 2014

A246616 Number of endofunctions on [n] whose cycle lengths are multiples of 9.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 40320, 3628800, 219542400, 11496038400, 570031862400, 27908676956160, 1379187810000000, 69648048665395200, 3619848890071814400, 194428045753727385600, 10817313676715756770560, 624148118196535296000000, 37365232440634029661766400

Offset: 0

Alois P. Heinz, Aug 31 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300

Column k=9 of A246609.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, add(b(n-i*j, i+9)*(i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
    end:
a:= a->add(b(j, 9)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);

CoefficientList[Series[1/(1+LambertW[-x]^9)^(1/9),{x,0,20}],x] * Range[0,20]!  (* Vaclav Kotesovec, Sep 01 2014 *)

E.g.f.: 1/(1+LambertW(-x)^9)^(1/9). - Vaclav Kotesovec, Sep 01 2014

a(n) ~ n^(n-4/9) * 2^(4/9) * sqrt(Pi) / (3^(2/9) * Gamma(1/18)) * (1 - 52 * sqrt(2/n) * Gamma(1/18) / (243 * Gamma(5/9))). - Vaclav Kotesovec, Sep 01 2014

A246617 Number of endofunctions on [n] whose cycle lengths are multiples of 10.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 362880, 39916800, 2874009600, 175394419200, 9967384627200, 551675124000000, 30471021291110400, 1703458301210265600, 97213825272736972800, 5693251850259515942400, 343266731083210449715200, 21349233350716392722764800

Offset: 0

Alois P. Heinz, Aug 31 2014

nonn

In general, column k of A246609 is (for k > 1) asymptotic to n^(n-1/2 + 1/(2*k)) * sqrt(2*Pi) / (2^(1/(2*k)) * k^(1/k) * Gamma(1/(2*k))) * (1 - (3*k-1)*(k-1) * sqrt(2/n) * Gamma(1/(2*k)) / (12 * k^2 * Gamma(1/2 + 1/(2*k)))). - Vaclav Kotesovec, Sep 01 2014

Alois P. Heinz, Table of n, a(n) for n = 0..300

Column k=10 of A246609.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, add(b(n-i*j, i+10)*(i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
    end:
a:= a->add(b(j, 10)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);

CoefficientList[Series[1/(1-LambertW[-x]^10)^(1/10),{x,0,20}],x] * Range[0,20]!  (* Vaclav Kotesovec, Sep 01 2014 *)

E.g.f.: 1/(1-LambertW(-x)^10)^(1/10). - Vaclav Kotesovec, Sep 01 2014

a(n) ~ n^(n-9/20) * 2^(7/20) * sqrt(Pi) / (5^(1/10) * Gamma(1/20)) * (1 - 87 * sqrt(2/n) * Gamma(1/20) / (400 * Gamma(11/20))). - Vaclav Kotesovec, Sep 01 2014

A246618 Number of endofunctions on [2n] whose cycle lengths are multiples of n.

Original entry on oeis.org

1, 4, 57, 4480, 866460, 302835456, 165589522560, 130247609057280, 139297568464454400, 194428045753727385600, 343266731083210449715200, 747889980460943180326502400, 1971026081420013638259189350400, 6180432779330984921337015828480000

Offset: 0

Alois P. Heinz, Aug 31 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..195

Cf. A246609.

with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i=0 or i>n, 0, add(b(n-i*j, i+k, k)*(i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
    end:
a:= n->add(b(j, n$2)*(2*n)^(2*n-j)*binomial(2*n-1, j-1), j=0..2*n):
seq(a(n), n=0..15);

multinomial[n_, k_] := n!/Times @@ (k!); b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i == 0 || i > n, 0, Sum[b[n - i*j, i + k, k]*(i - 1)!^j * multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!, {j, 0, n/i}]]]; a[0] = 1; a[n_] := Sum[b[j, n, n]*(2n)^(2n-j)*Binomial[2n-1, j-1], {j, 0, 2n}]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Feb 26 2017, after Alois P. Heinz *)

a(n) = A246609(2n,n).

a(n) ~ 2^(3*n-1/2) * n^(2*n-1) / exp(n). - Vaclav Kotesovec, Sep 01 2014

cp's OEIS Frontend

A060435 Number of functions f: {1,2,...,n} -> {1,2,...,n} with even cycles only.

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A246610 Number of endofunctions on [n] whose cycle lengths are multiples of 3.

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A246611 Number of endofunctions on [n] whose cycle lengths are multiples of 4.

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A246612 Number of endofunctions on [n] whose cycle lengths are multiples of 5.

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A246613 Number of endofunctions on [n] whose cycle lengths are multiples of 6.

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A246614 Number of endofunctions on [n] whose cycle lengths are multiples of 7.

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A246615 Number of endofunctions on [n] whose cycle lengths are multiples of 8.

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A246616 Number of endofunctions on [n] whose cycle lengths are multiples of 9.

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