A065440 -id:A065440 - cp's OEIS frontend

A007778 a(n) = n^(n+1).

0, 1, 8, 81, 1024, 15625, 279936, 5764801, 134217728, 3486784401, 100000000000, 3138428376721, 106993205379072, 3937376385699289, 155568095557812224, 6568408355712890625, 295147905179352825856, 14063084452067724991009, 708235345355337676357632

Offset: 0

N. J. A. Sloane

Number of edges of the complete bipartite graph of order n+n^n, K_n,n^n. - Roberto E. Martinez II, Jan 07 2002
All rational solutions to the equation x^y = y^x, with x < y, are given by x = A000169(n+1)/A000312(n), y = A000312(n+1)/A007778(n), where n >= 1. - Nick Hobson, Nov 30 2006
a(n) is also the number of ways of writing an n-cycle as the product of n+1 transpositions. - Nikos Apostolakis, Nov 22 2008
a(n) is the total number of elements whose preimage is the empty set summed over all partial functions from [n] into [n]. - Geoffrey Critzer, Jan 12 2022

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 67.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Nick Hobson, Exponential equation.
Yidong Sun and Jujuan Zhuang, lambda-factorials of n, arXiv:1007.1339 [math.CO], 2010. - _Peter Luschny_, Jul 09 2010

Cf. A000169, A000272, A000312, A007830, A008785, A008786, A008787, A008788, A008789, A008790, A008791, A135608.
Essentially the same as A065440.
Cf. A061250, A143857. [From Reinhard Zumkeller, Jul 23 2010]

[n^(n+1):n in [0..20]]; // Vincenzo Librandi, Jan 03 2012

seq( n^(n+1), n=0..20); # G. C. Greubel, Mar 05 2020

Table[n^(n+1), {n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Oct 01 2008 *)

A007778[n]:=n^(n+1)$
makelist(A007778[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

vector(21, n, my(m=n-1); m^(m+1)) \\ G. C. Greubel, Mar 05 2020

[n^(n+1) for n in (0..20)] # G. C. Greubel, Mar 05 2020

E.g.f.: -W(-x)/(1 + W(-x))^3, W(x) Lambert's function (principal branch).
a(n) = Sum_{k=0..n} binomial(n,k)*A000166(k+1)*(n+1)^(n-k). - Peter Luschny, Jul 09 2010
See A008517 and A134991 for similar e.g.f.s. and A048993. - Tom Copeland, Oct 03 2011
E.g.f.: d/dx {x/(T(x)*(1-T(x)))}, where T(x) = Sum_{n >= 1} n^(n-1)*x^n/n! is the tree function of A000169. - Peter Bala, Aug 05 2012
a(n) = n*A000312(n). - R. J. Mathar, Jan 12 2017
Sum_{n>=2} 1/a(n) = A135608. - Amiram Eldar, Nov 17 2020

A088956 Triangle, read by rows, of coefficients of the hyperbinomial transform.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 16, 9, 3, 1, 125, 64, 18, 4, 1, 1296, 625, 160, 30, 5, 1, 16807, 7776, 1875, 320, 45, 6, 1, 262144, 117649, 27216, 4375, 560, 63, 7, 1, 4782969, 2097152, 470596, 72576, 8750, 896, 84, 8, 1, 100000000, 43046721, 9437184, 1411788, 163296, 15750

Offset: 0

Paul D. Hanna, Oct 26 2003

The hyperbinomial transform of a sequence {b} is defined to be the sequence {d} given by d(n) = Sum_{k=0..n} T(n,k)*b(k), where T(n,k) = (n-k+1)^(n-k-1)*C(n,k).
Given a table in which the n-th row is the n-th binomial transform of the first row, then the hyperbinomial transform of any diagonal results in the next lower diagonal in the table.
The simplest example of a table of iterated binomial transforms is A009998, with a main diagonal of {1,2,9,64,625,...}; and the hyperbinomial transform of this diagonal gives the next lower diagonal, {1,3,16,125,1296,...}, since 1=(1)*1, 3=(1)*1+(1)*2, 16=(3)*1+(2)*2+(1)*9, 125=(16)*1+(9)*2+(3)*9+(1)*64, etc.
Another example: the hyperbinomial transform maps A065440 into A055541, since HYPERBINOMIAL([1,1,1,8,81,1024,15625]) = [1,2,6,36,320,3750,54432] where e.g.f.: A065440(x)+x = x-x/( LambertW(-x)*(1+LambertW(-x)) ), e.g.f.: A055541(x) = x-x*LambertW(-x).
The m-th iteration of the hyperbinomial transform is given by the triangle of coefficients defined by T_m(n,k) = m*(n-k+m)^(n-k-1)*binomial(n,k).
Example: PARI code for T_m: {a=[1,1,1,8,81,1024,15625]; m=1; b=vector(length(a)); for(n=0,length(a)-1, b[n+1]=sum(k=0,n, m*(n-k+m)^(n-k-1)*binomial(n,k)*a[k+1]); print1(b[n+1],","))} RETURNS b=[1,2,6,36,320,3750,54432].
The INVERSE hyperbinomial transform is thus given by m=-1: {a=[1,2,6,36,320,3750,54432]; m=-1; b=vector(length(a)); for(n=0,length(a)-1, b[n+1]=sum(k=0,n, m*(n-k+m)^(n-k-1)*binomial(n,k)*a[k+1]); print1(b[n+1],","))} RETURNS b=[1,1,1,8,81,1024,15625].
Simply stated, the HYPERBINOMIAL transform is to -LambertW(-x)/x as the BINOMIAL transform is to exp(x).
Let A[n] be the set of all forests of labeled rooted trees on n nodes. Build a superset B[n] of A[n] by designating "some" (possibly all or none) of the isolated nodes in each forest. T(n,k) is the number of elements in B[n] with exactly k designated nodes. See A219034. - Geoffrey Critzer, Nov 10 2012

			Rows begin:
       {1},
       {1,      1},
       {3,      2,     1},
      {16,      9,     3,    1},
     {125,     64,    18,    4,   1},
    {1296,    625,   160,   30,   5,  1},
   {16807,   7776,  1875,  320,  45,  6, 1},
  {262144, 117649, 27216, 4375, 560, 63, 7, 1}, ...

T. D. Noe, Rows n=0..50 of triangle, flattened
T. Copeland, Composition, Conjugation, and the Umbral Calculus-Part I, 2021.
G. Helms, Pascalmatrix tetrated
E. W. Weisstein, Abel Polynomial, From MathWorld - A Wolfram Web Resource.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A088957 (row sums), A000272 (first column), A009998, A105599, A132440, A215534 (matrix inverse), A215652.
Cf. A227325 (central terms).
Cf. A007318, A059297, A137542.

a088956 n k =  a095890 (n + 1) (k + 1) * a007318' n k `div` (n - k + 1)
a088956_row n = map (a088956 n) [0..n]
a088956_tabl = map a088956_row [0..]
-- Reinhard Zumkeller, Jul 07 2013

nn=8; t=Sum[n^(n-1)x^n/n!, {n,1,nn}]; Range[0,nn]! CoefficientList[Series[Exp[t+y x] ,{x,0,nn}], {x,y}] //Grid (* Geoffrey Critzer, Nov 10 2012 *)

T(n, k) = (n-k+1)^(n-k-1)*C(n, k).
E.g.f.: -LambertW(-x)*exp(x*y)/x. - Vladeta Jovovic, Oct 27 2003
From Peter Bala, Sep 11 2012: (Start)
Let T(x) = Sum_{n >= 0} n^(n-1)*x^n/n! denote the tree function of A000169. The e.g.f. is (T(x)/x)*exp(t*x) = exp(T(x))*exp(t*x) = 1 + (1 + t)*x + (3 + 2*t + t^2)*x^2/2! + .... Hence the triangle is the exponential Riordan array [T(x)/x,x] belonging to the exponential Appell group.
The matrix power (A088956)^r has the e.g.f. exp(r*T(x))*exp(t*x) with triangle entries given by r*(n-k+r)^(n-k-1)*binomial(n,k) for n and k >= 0. See A215534 for the case r = -1.
Let A(n,x) = x*(x+n)^(n-1) be an Abel polynomial. The present triangle is the triangle of connection constants expressing A(n,x+1) as a linear combination of the basis polynomials A(k,x), 0 <= k <= n. For example, A(4,x+1) = 125*A(0,x) + 64*A(1,x) + 18*A(2,x) + 4*A(3,x) + A(4,x) gives row 4 as [125,64,18,4,1].
Let S be the array with the sequence [1,2,3,...] on the main subdiagonal and zeros elsewhere. S is the infinitesimal generator for Pascal's triangle (see A132440). Then the infinitesimal generator for this triangle is S*A088956; that is, A088956 = Exp(S*A088956), where Exp is the matrix exponential.
With T(x) the tree function as above, define E(x) = T(x)/x. Then A088956 = E(S) = Sum_{n>=0} (n+1)^(n-1)*S^n/n!.
For commuting lower unit triangular matrices A and B, we define A raised to the matrix power B, denoted A^^B, to be the matrix Exp(B*log(A)), where the matrix logarithm Log(A) is defined as Sum_{n >= 1} (-1)^(n+1)*(A-1)^n/n. Let P denote Pascal's triangle A007318. Then the present triangle, call it X, solves the matrix equation P^^X = X . See A215652 for the solution to X^^P = P. Furthermore, if we denote the inverse of X by Y then X^^Y = P. As an infinite tower of matrix powers, A088956 = P^^(P^^(P^^(...))).
A088956 augmented with the sequence (x,x,x,...) on the first superdiagonal is the production matrix for the row polynomials of A105599.
(End)
T(n,k) = A095890(n+1,k+1) * A007318(n,k) / (n-k+1), 0 <= k <= n. - Reinhard Zumkeller, Jul 07 2013
Sum_{k = 0..n} T(n,n-k)*(x - k - 1)^(n-k) = x^n. Setting x = n + 1 gives Sum_{k = 0..n} T(n,k)*k^k = (n + 1)^n. - Peter Bala, Feb 17 2017
As lower triangular matrices, this entry, T, equals unsigned A137542 * A007318 * signed A059297. The Pascal matrix is sandwiched between a pair of inverse matrices, so this entry is conjugate to the Pascal matrix, allowing convergent analytic expressions of T, say f(T), to be computed as f(A007318) sandwiched between the inverse pair. - Tom Copeland, Dec 06 2021

A274804 The exponential transform of sigma(n).

Original entry on oeis.org

1, 1, 4, 14, 69, 367, 2284, 15430, 115146, 924555, 7991892, 73547322, 718621516, 7410375897, 80405501540, 914492881330, 10873902417225, 134808633318271, 1738734267608613, 23282225008741565, 323082222240744379, 4638440974576329923, 68794595993688306903

Offset: 0

Johannes W. Meijer, Jul 27 2016

nonn

The exponential transform [EXP] transforms an input sequence b(n) into the output sequence a(n). The EXP transform is the inverse of the logarithmic transform [LOG], see the Weisstein link and the Sloane and Plouffe reference. This relation goes by the name of Riddell's formula. For information about the logarithmic transform see A274805. The EXP transform is related to the multinomial transform, see A274760 and the second formula.
The definition of the EXP transform, see the second formula, shows that n >= 1. To preserve the identity LOG[EXP[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the exponential transform, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the multinomial coefficients A178867 appear.
We observe that a(0) = 1 and provides no information about any value of b(n), this notwithstanding it is customary to start the a(n) sequence with a(0) = 1.
The Maple programs can be used to generate the exponential transform of a sequence. The first program uses a formula found by Alois P. Heinz, see A007446 and the first formula. The second program uses the definition of the exponential transform, see the Weisstein link and the second formula. The third program uses information about the inverse of the exponential transform, see A274805.
Some EXP transform pairs are, n >= 1: A000435(n) and A065440(n-1); 1/A000027(n) and A177208(n-1)/A177209(n-1); A000670(n) and A075729(n-1); A000670(n-1) and A014304(n-1); A000045(n) and A256180(n-1); A000290(n) and A033462(n-1); A006125(n) and A197505(n-1); A053549(n) and A198046(n-1); A000311(n) and A006351(n); A030019(n) and A134954(n-1); A038048(n) and A053529(n-1); A193356(n) and A003727(n-1).

			Some a(n) formulas, see A178867:
a(0) = 1
a(1) = x(1)
a(2) = x(1)^2 + x(2)
a(3) = x(1)^3 + 3*x(1)*x(2) + x(3)
a(4) = x(1)^4 + 6*x(1)^2*x(2) + 4*x(1)*x(3) + 3*x(2)^2 + x(4)
a(5) = x(1)^5 + 10*x(1)^3*x(2) + 10*x(1)^2*x(3) + 15*x(1)*x(2)^2 + 5*x(1)*x(4) + 10*x(2)*x(3) + x(5)

Frank Harary and Edgar M. Palmer, Graphical Enumeration, 1973.
Robert James Riddell, Contributions to the theory of condensation, Dissertation, University of Michigan, Ann Arbor, 1951.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

Alois P. Heinz, Table of n, a(n) for n = 0..531
M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers, Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Exponential Transform.

Cf. A274805, A274760, A178867, A000203, A007446.
Cf. A177208, A177209, A006351, A197505, A144180, A256180, A033462, A198046, A134954, A145460, A188489, A005432, A029725, A124213, A002801.
Cf. A036040, another version.

nmax:=21: with(numtheory): b := proc(n): sigma(n) end: a:= proc(n) option remember; if n=0 then 1 else add(binomial(n-1, j-1) * b(j) *a(n-j), j=1..n) fi: end: seq(a(n), n=0..nmax); # End first EXP program.
nmax:= 21: with(numtheory): b := proc(n): sigma(n) end: t1 := exp(add(b(n)*x^n/n!, n=1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n!*coeff(t2, x, n) end: seq(a(n), n=0..nmax); # End second EXP program.
nmax:=21: with(numtheory): b := proc(n): sigma(n) end: f := series(log(1+add(q(n)*x^n/n!, n=1..nmax+1)), x, nmax+1): d := proc(n): n!*coeff(f, x, n) end: a(0):=1: q(0):=1: a(1):=b(1): q(1):=b(1): for n from 2 to nmax+1 do q(n) := solve(d(n)-b(n), q(n)): a(n):=q(n): od: seq(a(n), n=0..nmax); # End third EXP program.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n-1, j-1]*DivisorSigma[1, j]*a[n-j], {j, 1, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 22 2017 *)
nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k]*x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 08 2021 *)

a(n) = Sum_{j=1..n} (binomial(n-1,j-1) * b(j) * a(n-j)), n >= 1 and a(0) = 1, with b(n) = A000203(n) = sigma(n).

E.g.f.: exp(Sum_{n >= 1} b(n)*x^n/n!) with b(n) = sigma(n) = A000203(n).

A302583 a(n) = ((n + 1)^n - (n - 1)^n)/2.

Original entry on oeis.org

0, 1, 4, 28, 272, 3376, 51012, 908608, 18640960, 432891136, 11225320100, 321504185344, 10079828372880, 343360783937536, 12627774819845668, 498676704524517376, 21046391759976988928, 945381827279671853056, 45032132922921758270916, 2267322327322331161821184

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

nonn

N. J. A. Sloane, Transforms

Cf. A000169, A065440, A007778, A062024, A115416, A274278, A293022, A302584, A302585, A302586, A302587.

Table[((n + 1)^n - (n - 1)^n)/2, {n, 0, 19}]
nmax = 19; CoefficientList[Series[(x^2 - LambertW[-x]^2)/(2 x LambertW[-x] (1 + LambertW[-x])), {x, 0, nmax}], x] Range[0, nmax]!
Table[n! SeriesCoefficient[Exp[n x] Sinh[x], {x, 0, n}], {n, 0, 19}]

E.g.f.: (x^2 - LambertW(-x)^2)/(2*x*LambertW(-x)*(1 + LambertW(-x))).

a(n) = n! * [x^n] exp(n*x)*sinh(x).

A055134 Triangle read by rows: T(n,k) = number of labeled endofunctions on n points with k fixed points.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 8, 12, 6, 1, 81, 108, 54, 12, 1, 1024, 1280, 640, 160, 20, 1, 15625, 18750, 9375, 2500, 375, 30, 1, 279936, 326592, 163296, 45360, 7560, 756, 42, 1, 5764801, 6588344, 3294172, 941192, 168070, 19208, 1372, 56, 1, 134217728

Offset: 0

Christian G. Bower, Apr 25 2000

The same triangle (except for signs) may be obtained from the determinants of the Brahmagupta matrices, setting x->Sqrt[z], y->1, t->n. - Roger L. Bagula, Apr 09 2008
From Bob Selcoe, Nov 15 2014 (Start):
T(n,k)/A000312(n) is the probability P(n,k) that any member (j) of set J={1..n} will be selected k times given n random draws from J. This is equivalent to rolling an n-sided die (with standard assumptions) with sides numbered j=1..n: P(n,k) is the probability that any j will show k times with n rolls.
P(n,k) = (n-2)!*(n-1)^(n-k+1 )/k!*(n-k)!*n^(n-1); n>1. As n approaches infinity, P(n,0) and P(n,1) approach 1/e. (End)
Row sums give n^n (see A000312). - Bob Selcoe, Sep 08 2015

			Triangle T(n,k) begins:
       1;
       0,      1;
       1,      2,      1;
       8,     12,      6,     1;
      81,    108,     54,    12,    1;
    1024,   1280,    640,   160,   20,   1;
   15625,  18750,   9375,  2500,  375,  30,  1;
  279936, 326592, 163296, 45360, 7560, 756, 42, 1;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Eric W. Weisstein's World of Mathematics, Brahmagupta Matrix.

Columns k=0-2 give: A065440, A055897, A081132(n-2) for n>=2.
Row sums give A000312.
Cf. A055135, A055136.

Clear[B] B[0] = {{x, y}, {t*y, x}}; B[n_] := B[n] = B[n - 1].B[0]; Table[Det[B[n]] /. x -> Sqrt[z] /. y -> 1 /. t -> n, {n, 0, 10}]; a = Join[{{1}}, Table[CoefficientList[Det[B[n]] /. x -> Sqrt[z] /. y ->1 /. t -> n, z], {n, 0, 10}]]; Flatten[a] (* Roger L. Bagula, Apr 09 2008 *)
row[n_] := CoefficientList[(x + n - 1)^n + O[x]^(n+1), x];
Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 13 2017, after Geoffrey Critzer *)
Join[{1, 0, 1}, Table[Binomial[n, k]*(n - 1)^(n - k), {n, 2, 49}, {k, 0, n}]] // Flatten (* G. C. Greubel, Nov 14 2017 *)

for(n=0,15, for(k=0,n, print1(if(n==0 && k==0, 1, if(n==1 && k==0, 0, if(n==1 && k==1, 1, binomial(n,k)*(n-1)^(n-k)))), ", "))) \\ G. C. Greubel, Nov 14 2017

T(n, k) = C(n, k)*(n-1)^(n-k), for n>1.
E.g.f.: (-LambertW(-y)/y)^(x-1)/(1+LambertW(-y)). - Vladeta Jovovic
O.g.f. for row n: (x + n - 1)^n. - Geoffrey Critzer, Mar 21 2010

A085606 a(n) = (n-1)^n - 1.

Original entry on oeis.org

0, -1, 0, 7, 80, 1023, 15624, 279935, 5764800, 134217727, 3486784400, 99999999999, 3138428376720, 106993205379071, 3937376385699288, 155568095557812223, 6568408355712890624, 295147905179352825855, 14063084452067724991008, 708235345355337676357631

Offset: 0

Lekraj Beedassy, Jul 07 2003

sign

Sequence relates to the "monkey and coconut problem"(A014293) giving the number of coconuts received by each of the n sailors from the ultimate equitable distribution the next day.
From Alexander Adamchuk, Nov 13 2006: (Start)
4n^2 divides a(2n).
Odd prime p divides a(p-1).
8p^2 divides a(2p) for an odd prime p.
32p^4 divides a(2p^2) for an odd prime p.
64p^8 divides a(2p^4) for an odd prime p.
p^3 divides a(p^3+2) for prime p.
p divides a((p-1)/2) for prime p in A157437.
p^2 divides a((p-1)/2) for prime p = {5,127,607}. (End)

Cf. A014293, A007778, A065440, A037205, A157437.

Table[(n-1)^n-1,{n,0,30}] (* Alexander Adamchuk, Nov 13 2006 *)

a(n) = A065440(n) - 1.

More terms from Ray Chandler, Nov 10 2003

A284458 Number of pairs (f,g) of endofunctions on [n] such that the composite function gf has no fixed point.

Original entry on oeis.org

1, 0, 2, 156, 16920, 2764880, 650696400, 210105425628, 89425255439744, 48588905856409920, 32845298636854828800, 27047610425293718239100, 26664178085975252011318272, 31009985808408471580603417296, 42017027730087624384021945067520

Offset: 0

Robert FERREOL, Mar 27 2017

nonn

Consider n boys and n girls, each boy secretly loving a girl and vice versa. The probability that no couple could be formed is a(n)/n^(2n).
a(n) is the number of pairs of binary matrices n X n (M, N), M having only one 1 per row, N having only one 1 per column, such that M + N has no coefficient equal to 2.
Limit_{n -> infinity} a(n)/n^(2n) = 1/e.

			For two boys A,B and two girls A',B', the a(2) possibilities are:
A loves A' who loves B who loves B' who loves A;
A loves B' who loves B who loves A' who loves A.

Robert Israel, Table of n, a(n) for n = 0..214
Math StackExchange, An unexpected application of non-trivial combinatorics, 2014.
Math StackExchange, Couple Probability, 2014.
Eric Weisstein's MathWorld, Confluent Hypergeometric Function of the Second Kind

Cf. A065440, A350558, A343700.

a:=n->add((-1)^k*binomial(n,k)^2*k!*n^(2*(n-k)),k=0..n):
seq(a(n),n=0..15);

Table[Sum[(-1)^k Binomial[n,k]^2 * k! * n^(2*(n - k)), {k, 0, n}], {n, 1, 15}] (* Indranil Ghosh, Mar 27 2017 *)
Table[HypergeometricU[-n, 1, n^2], {n, 1, 15}] (* Jean-François Alcover, Mar 29 2017 *)

a(n) = sum(k=0, n, (-1)^k * binomial(n,k)^2 * k! * n^(2*(n-k))); \\ Michel Marcus, Apr 04 2017

a(n) = Sum_{k=0..n} (-1)^k * binomial(n,k)^2 * k! * n^(2*(n-k)).

a(n) = A343700(n)/A350558(n). - Dan Eilers, Jan 17 2023

A349454 Number T(n,k) of endofunctions on [n] with exactly k fixed points, all of which are isolated; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 8, 3, 0, 1, 81, 32, 6, 0, 1, 1024, 405, 80, 10, 0, 1, 15625, 6144, 1215, 160, 15, 0, 1, 279936, 109375, 21504, 2835, 280, 21, 0, 1, 5764801, 2239488, 437500, 57344, 5670, 448, 28, 0, 1, 134217728, 51883209, 10077696, 1312500, 129024, 10206, 672, 36, 0, 1

Offset: 0

Alois P. Heinz, Dec 30 2021

			Triangle T(n,k) begins:
        1;
        0,       1;
        1,       0,      1;
        8,       3,      0,     1;
       81,      32,      6,     0,    1;
     1024,     405,     80,    10,    0,   1;
    15625,    6144,   1215,   160,   15,   0,  1;
   279936,  109375,  21504,  2835,  280,  21,  0, 1;
  5764801, 2239488, 437500, 57344, 5670, 448, 28, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Column k=0 gives A065440.
Row sums give A204042.
Main diagonal and first lower diagonal give A000012, A000004.
T(n+1,n-1) gives A000217.
T(n+3,n) gives A130809.
T(n+3,n-1) gives A102741 for n>=1.
Cf. A055134, A217701, A350212, A350454.

T:= (n, k)-> binomial(n, k)*(n-k-1)^(n-k):
seq(seq(T(n, k), k=0..n), n=0..10);

T(n,k) = binomial(n,k) * (n-k-1)^(n-k).
From Mélika Tebni, Apr 02 2023: (Start)
E.g.f. of column k: -x / (LambertW(-x)*(1+LambertW(-x)))*x^k / k!.
Sum_{k=0..n} k^k*T(n,k) = A217701(n). (End)

A134362 a(n) is the number of functions f:X->X, where |X| = n, such that for every x in X, f(f(x)) != x (i.e., the square of the function has no fixed points; note this implies that the function has no fixed points).

Original entry on oeis.org

1, 0, 0, 2, 30, 444, 7360, 138690, 2954364, 70469000, 1864204416, 54224221050, 1721080885480, 59217131089908, 2195990208122880, 87329597612123594, 3707783109757616400, 167411012044894728720, 8010372386879991018496, 404912918159552083622130

Offset: 0

Adam Day (adam.r.day(AT)gmail.com), Jan 17 2008

nonn

This sequence arose when analyzing the Zen Stare game. This game is played with a group of people standing in a circle. They start heads bowed and then everyone raises their heads simultaneously and looks at someone else in the circle. If no two people are looking at each other a Zen Stare is achieved.

			a(3) = 2 because given a three-element set X:= {A, B, C} the only functions whose square has no fixed points are f:X->X where f(A)=B, f(B)=C, f(C)=A and g:X->X where g(A)=C, g(B)=A, g(C)=B.

Mohammad K. Azarian, On the Fixed Points of a Function and the Fixed Points of its Composite Functions, International Journal of Pure and Applied Mathematics, Vol. 46, No. 1, 2008, pp. 37-44. Mathematical Reviews, MR2433713 (2009c:65129), March 2009. Zentralblatt MATH, Zbl 1160.65015.
Mohammad K. Azarian, Fixed Points of a Quadratic Polynomial, Problem 841, College Mathematics Journal, Vol. 38, No. 1, January 2007, p. 60. Solution published in Vol. 39, No. 1, January 2008, pp. 66-67.

Alois P. Heinz, Table of n, a(n) for n = 0..150
P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009, page 130.
Marko Riedel, Proof of EGF and closed form

Cf. A065440, A007778.

a:= n -> (n-1)^n + add((-1)^i*mul(binomial(n-2*(j-1),2),j=1..i)*(n-1)^(n-2*i)/i!,i=1..floor(n/2)): seq(a(n), n=0..20);

nn = 20; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}]; Drop[Range[0, nn]! CoefficientList[Series[Exp[-t - t^2/2]/(1 - t), {x, 0, nn}], x], 1] (* Geoffrey Critzer, Feb 06 2012 *)

a(n) = n!*sum(q=0, n\2, ((-1)^q/(2^q*q!)*(n-1)^(n-2*q)/(n-2*q)!)); \\ Michel Marcus, Mar 09 2016

E.g.f.: exp(-T(x)-T(x)^2/2)/(1-T(x)) where T(x) is the e.g.f. for A000169. - Geoffrey Critzer, Feb 06 2012
a(n) ~ exp(-3/2) * n^n. - Vaclav Kotesovec, Aug 16 2013
a(n) = n!*Sum_{q=0..floor(n/2)} ((-1)^q/(2^q q!) * (n-1)^(n-2q)/(n-2q)!). - Marko Riedel, Mar 08 2016

A204042 The number of functions f:{1,2,...,n}->{1,2,...,n} (endofunctions) such that all of the fixed points in f are isolated.

Original entry on oeis.org

1, 1, 2, 12, 120, 1520, 23160, 413952, 8505280, 197631072, 5125527360, 146787894440, 4601174623584, 156693888150384, 5761055539858528, 227438694372072120, 9596077520725211520, 430920897407809702208, 20520683482765477749120, 1032920864149903149579336, 54797532208320308334631840

Offset: 0

Geoffrey Critzer, Jan 09 2012

nonn

Note this sequence counts the functions enumerated by A065440 for which the statement is vacuously true.

a(n) is also the number of partial endofunctions on {1,2,...,n} without fixed points.

			a(2)=2 because there are two functions f:{1,2}->{1,2} in which all the fixed points are isolated: 1->1,2->2  and 1->2,2->1 (which has no fixed points).

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

Cf. A065440, A086331, A350134.

Row sums of A349454.

a:= n-> add((j-1)^j*binomial(n, j), j=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Dec 16 2021

t = Sum[n^(n-1) x^n/n!, {n,1,20}]; Range[0,20]! CoefficientList[Series[Exp[x] Exp[Log[1/(1-t)]-t], {x,0,20}], x]

E.g.f.: exp(x)*A(x) where A(x) is the e.g.f. for A065440.
a(n) ~ exp(exp(-1)-1)*n^n. - Vaclav Kotesovec, Sep 24 2013
a(n) = Sum_{j=0..n} (j-1)^j * binomial(n,j). - Alois P. Heinz, Dec 16 2021

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