A060281 -id:A060281 - cp's OEIS frontend

A350452 Number T(n,k) of endofunctions on [n] with exactly k connected components and no fixed points; triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

1, 0, 0, 1, 0, 8, 0, 78, 3, 0, 944, 80, 0, 13800, 1810, 15, 0, 237432, 41664, 840, 0, 4708144, 1022252, 34300, 105, 0, 105822432, 27098784, 1286432, 10080, 0, 2660215680, 778128336, 47790540, 648900, 945, 0, 73983185000, 24165049920, 1815578160, 36048320, 138600

Offset: 0

Alois P. Heinz, Dec 31 2021

For k >= 2 and p prime, T(p,k) == 0 (mod 4*p*(p-1)). - Mélika Tebni, Jan 20 2023

			Triangle T(n,k) begins:
  1;
  0;
  0,          1;
  0,          8;
  0,         78,         3;
  0,        944,        80;
  0,      13800,      1810,       15;
  0,     237432,     41664,      840;
  0,    4708144,   1022252,    34300,    105;
  0,  105822432,  27098784,  1286432,  10080;
  0, 2660215680, 778128336, 47790540, 648900, 945;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-1 give: A000007, A000435.
Row sums give A065440.
T(2n,n) gives A001147.
Cf. A060281, A061356, A350446.
Cf. A071720, A007830, A106828.

c:= proc(n) option remember; add(n!*n^(n-k-1)/(n-k)!, k=2..n) end:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(
      b(n-i)*binomial(n-1, i-1)*x*c(i), i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n)):
seq(T(n), n=0..12);

c[n_] := c[n] = Sum[n!*n^(n - k - 1)/(n - k)!, {k, 2, n}];
b[n_] := b[n] = Expand[If[n == 0, 1, Sum[
     b[n - i]*Binomial[n - 1, i - 1]*x*c[i], {i, 1, n}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n/2}]][b[n]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Mar 18 2022, after Alois P. Heinz *)

\\ here AS1(n,k) gives associated Stirling numbers of 1st kind.
AS1(n,k)={(-1)^(n+k)*sum(i=0, k, (-1)^i * binomial(n, i) * stirling(n-i, k-i, 1) )}
T(n,k) = {if(n==0, k==0, sum(j=k, n, n^(n-j)*binomial(n-1, j-1)*AS1(j,k)))} \\ Andrew Howroyd, Jan 20 2023

From Mélika Tebni, Jan 20 2023: (Start)
E.g.f. column k: (LambertW(-x) - log(1 + LambertW(-x)))^k / k!.
-Sum_{k=1..n/2} (-1)^k*T(n,k) = A071720(n+1), for n > 0.
-Sum_{k=1..n/2} (-1)^k*T(n,k) / (n-1) = A007830(n-2), for n > 1.
T(n,k) = Sum_{j=k..n} n^(n-j)*binomial(n-1, j-1)*A106828(j, k) for n > 0. (End)

A273434 Number of endofunctions on [n] with exactly three cycles.

Original entry on oeis.org

1, 18, 305, 5595, 113974, 2581964, 64727522, 1783995060, 53705023251, 1755078270264, 61920105083187, 2346728722199680, 95117694573257784, 4106779625155078528, 188206877039146217476, 9125798298446360109312, 466820173490890114763781, 25126459591455539907002880

Offset: 3

Alois P. Heinz, May 22 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..386

Column k=3 of A060281.

Drop[CoefficientList[Series[-1/6 * Log[1+LambertW[-x]]^3, {x, 0, 20}], x] * Range[0, 20]!, 3] (* Vaclav Kotesovec, Nov 01 2016 *)

x='x+O('x^30); Vec(serlaplace(-log(1+lambertw(-x))^3/6)) \\ G. C. Greubel, Aug 30 2018

E.g.f.: -1/6 * log(1+LambertW(-x))^3.

a(n) ~ n^(n-1/2) * sqrt(2*Pi) * (log(n))^2 / 16 * (1 + 2*(gamma - log(2))/log(n) + (gamma^2 - 2*log(2)*gamma + log(2)^2 - Pi^2/6)/log(n)^2), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Nov 01 2016

A273435 Number of endofunctions on [n] with exactly four cycles.

Original entry on oeis.org

1, 30, 745, 18515, 484729, 13591116, 409987640, 13303809750, 463397746636, 17276343754098, 687247936771032, 29079485483123985, 1304889365985201424, 61922948839969015928, 3099416199490785094272, 163229892984351540698188, 9024648860521246301700096

Offset: 4

Alois P. Heinz, May 22 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..386

Column k=4 of A060281.

E.g.f.: 1/24 * log(1+LambertW(-x))^4.

A273436 Number of endofunctions on [n] with exactly five cycles.

Original entry on oeis.org

1, 45, 1540, 49840, 1632099, 55545735, 1987186025, 75078221130, 2999597292406, 126693681294180, 5650573288138415, 265702055516788800, 13149171975158028874, 683615652343279677360, 37269087381803600233878, 2126880663709734887508320, 126841125623724152774643951

Offset: 5

Alois P. Heinz, May 22 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..386

Column k=5 of A060281.

E.g.f.: -1/5! * log(1+LambertW(-x))^5.

A273437 Number of endofunctions on [n] with exactly six cycles.

Original entry on oeis.org

1, 63, 2842, 116172, 4654713, 189142107, 7923937307, 345104368752, 15688849238062, 745831503236820, 37094603885430728, 1929672890969261256, 104918114960824458448, 5956513043619244790970, 352719666690509340493680, 21759830035878816514854144

Offset: 6

Alois P. Heinz, May 22 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..386

Column k=6 of A060281.

E.g.f.: 1/6! * log(1+LambertW(-x))^6.

A273438 Number of endofunctions on [n] with exactly seven cycles.

Original entry on oeis.org

1, 84, 4830, 243390, 11717013, 560544138, 27196758875, 1353285904971, 69495472079033, 3696068344472504, 203958595104203576, 11687491140975605592, 695597351310503327386, 42988755956609918306640, 2757417607812192585058358, 183451189952939198906121968

Offset: 7

Alois P. Heinz, May 22 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..386

Column k=7 of A060281.

E.g.f.: -1/7! * log(1+LambertW(-x))^7.

A273439 Number of endofunctions on [n] with exactly eight cycles.

Original entry on oeis.org

1, 108, 7710, 469590, 26750823, 1488656598, 82839614429, 4672159568505, 269240651261153, 15931093968844912, 970803214965494976, 61032579212845013280, 3962346372673358180536, 265758042459395190801852, 18415788640437877637900864, 1318210727391112435230475378

Offset: 8

Alois P. Heinz, May 22 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..386

Column k=8 of A060281.

E.g.f.: 1/8! * log(1+LambertW(-x))^8.

A273440 Number of endofunctions on [n] with exactly nine cycles.

Original entry on oeis.org

1, 135, 11715, 848430, 56457258, 3616660047, 228859354295, 14526558274800, 933767559522083, 61157779792168059, 4097522595384976251, 281552808746124677190, 19873435970688901729621, 1442409484607029199323392, 107706418622406296313956423

Offset: 9

Alois P. Heinz, May 22 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..386

Column k=9 of A060281.

E.g.f.: -1/9! * log(1+LambertW(-x))^9.

A273441 Number of endofunctions on [n] with exactly ten cycles.

Original entry on oeis.org

1, 165, 17105, 1452880, 111684573, 8161132980, 582816362700, 41367826311240, 2950036744905393, 212871288926657075, 15617503320899550135, 1168755168529889495100, 89415279382066253241846, 7003566076056061232032785, 562166804049332506124053492

Offset: 10

Alois P. Heinz, May 22 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..386

Column k=10 of A060281.

E.g.f.: 1/10! * log(1+LambertW(-x))^10.

A273442 Number of endofunctions on [2n] with exactly n cycles.

Original entry on oeis.org

1, 3, 95, 5595, 484729, 55545735, 7923937307, 1353285904971, 269240651261153, 61157779792168059, 15617503320899550135, 4429016799173481942427, 1381112305978592892946825, 469689278931628969590283855, 173002815169302537782725771395

Offset: 0

Alois P. Heinz, May 22 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..288

Cf. A060281.

Table[(2*n)!/n! * SeriesCoefficient[(-Log[1+LambertW[-x]])^n, {x, 0, 2*n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 01 2016 *)
Flatten[{1, Table[Sum[Binomial[2*n-1, k] * (2*n)^(2*n-1-k) * Abs[StirlingS1[k+1, n]], {k, 0, 2*n-1}], {n, 1, 20}]}] (* Vaclav Kotesovec, Nov 01 2016 *)

a(n) = (2*n)!/n! * [x^(2*n)] (-log(1+LambertW(-x)))^n.
a(n) = A060281(2n,n).
a(n) ~ c * d^n * n^(n-1/2), where d = 2^(4-r) * exp(1-r) * (2-r)^(r-2) * log(s) / (1-1/s)^r = 10.40858458700790823344027277763248832..., where r = 1.2672171362228848078038115564503589940694831794020694762759870935... is the root of the equation r*log(s) * (-1 + (r-s)* log((2*(s-1))/(s*(2-r)))) = 1 - s, where s = -r*LambertW(-1, -exp(-1/r)/r) = 1.5782614856055967129193228312616913... and c = 0.336740238865974324583136447665761... - Vaclav Kotesovec, Nov 01 2016, extended Aug 28 2017

cp's OEIS Frontend

A350452 Number T(n,k) of endofunctions on [n] with exactly k connected components and no fixed points; triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A273434 Number of endofunctions on [n] with exactly three cycles.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A273435 Number of endofunctions on [n] with exactly four cycles.

Views

Author

Keywords

Links

Crossrefs

Formula

A273436 Number of endofunctions on [n] with exactly five cycles.

Views

Author

Keywords

Links

Crossrefs

Formula

A273437 Number of endofunctions on [n] with exactly six cycles.

Views

Author

Keywords

Links

Crossrefs

Formula

A273438 Number of endofunctions on [n] with exactly seven cycles.

Views

Author

Keywords

Links

Crossrefs

Formula

A273439 Number of endofunctions on [n] with exactly eight cycles.

Views

Author

Keywords

Links

Crossrefs

Formula

A273440 Number of endofunctions on [n] with exactly nine cycles.

Views

Author

Keywords

Links

Crossrefs

Formula

A273441 Number of endofunctions on [n] with exactly ten cycles.

Views

Author

Keywords

Links

Crossrefs

Formula

A273442 Number of endofunctions on [2n] with exactly n cycles.

Views

Author

Keywords

Links

Crossrefs

Programs