A245910 -id:A245910 - cp's OEIS frontend

A062206 a(n) = n^(2n).

1, 1, 16, 729, 65536, 9765625, 2176782336, 678223072849, 281474976710656, 150094635296999121, 100000000000000000000, 81402749386839761113321, 79496847203390844133441536, 91733330193268616658399616009, 123476695691247935826229781856256

Offset: 0

Jason Earls, Jun 13 2001

a(n) is also the number of sequences of length 2n on n symbols. - Washington Bomfim, Oct 06 2009

a(n) is the number of endofunctions on [n] that map each even number to an even number and each odd number to an odd number. - Enrique Navarrete, Sep 30 2022

Winston de Greef, Table of n, a(n) for n = 0..213 (first 101 terms from Harry J. Smith)

Cf. A062207, A086648, A155957.
Cf. A000290, A000312, A155955.
Cf. A085741, A212333.
Column k=0 of A245910 and A245980.

f[n_]:=n^(2*n); Join[{1},f[Range[20]]] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2011 *)

a(n) = n^(2*n); \\ Harry J. Smith, Aug 02 2009

def A062206(n): return n**(n<<1) # Chai Wah Wu, Nov 18 2022

a(n) = A000312(n)^2 = A000290(n)^n.
(-1)^n*determinant of the 2n X 2n matrix M_(i, j) = i+j if (i + j) is a multiple of n, M_(i, j) = 1 otherwise. - Benoit Cloitre, Aug 06 2003
a(n) = A155955(n,n) = A000290(A000312(n)). - Reinhard Zumkeller, Jan 31 2009
a(n) = n! * [x^n] 1/(1 + LambertW(-n*x)). - Ilya Gutkovskiy, Oct 03 2017
Sum_{n>=1} 1/a(n) = A086648. - Amiram Eldar, Nov 16 2020

Initial term corrected by Reinhard Zumkeller, Jan 30 2009

A245980 Number A(n,k) of pairs of endofunctions f, g on [n] satisfying g^k(f(i)) = f(i) for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 16, 1, 1, 6, 729, 1, 1, 10, 87, 65536, 1, 1, 6, 213, 2200, 9765625, 1, 1, 10, 141, 8056, 84245, 2176782336, 1, 1, 6, 213, 6184, 465945, 4492656, 678223072849, 1, 1, 10, 87, 9592, 387545, 37823616, 315937195, 281474976710656

Offset: 0

Alois P. Heinz, Aug 08 2014

			Square array A(n,k) begins:
0 :        1,     1,      1,      1,      1,      1, ...
1 :        1,     1,      1,      1,      1,      1, ...
2 :       16,     6,     10,      6,     10,      6, ...
3 :      729,    87,    213,    141,    213,     87, ...
4 :    65536,  2200,   8056,   6184,   9592,   2200, ...
5 :  9765625, 84245, 465945, 387545, 682545, 159245, ...

Alois P. Heinz, Antidiagonals n = 0..80, flattened

Columns k=0-10 give: A062206, A239750, A239771, A241015, A245981, A245982, A245983, A245984, A245985, A245986, A245987.
Main diagonal gives A245988.
Cf. A245910.

with(numtheory): with(combinat): M:=multinomial:
b:= proc(n, k, p) local l, g; l, g:= sort([divisors(p)[]]),
      proc(k, m, i, t) option remember; local d, j; d:= l[i];
        `if`(i=1, n^m, add(M(k, k-(d-t)*j, (d-t)$j)/j!*
         (d-1)!^j *M(m, m-t*j, t$j) *g(k-(d-t)*j, m-t*j,
        `if`(d-t=1, [i-1, 0], [i, t+1])[]), j=0..min(k/(d-t),
        `if`(t=0, [][], m/t))))
      end; g(k, n-k, nops(l), 0)
    end:
A:= (n, k)-> `if`(k=0, n^(2*n), add(b(n, j, k)*
             stirling2(n, j)*binomial(n, j)*j!, j=0..n)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

multinomial[n_, k_List] := n!/Times @@ (k!); M = multinomial;
b[n_, k0_, p_] := Module[{l, g}, l = Sort[Divisors[p]]; g[k_, m_, i_, t_] := g[k, m, i, t] = Module[{d, j}, d = l[[i]]; If[i == 1, n^m, Sum[M[k, Join[{k - (d-t)*j}, Array[(d - t)&, j]]]/ j!*(d-1)!^j * M[m, Join[{m - t*j}, Array[t&, j]]]*If[d-t == 1, g[k - (d - t)*j, m - t*j, i-1, 0], g[k - (d-t)*j, m - t*j, i, t+1]], {j, 0, Min[k/(d-t), If[t == 0, Infinity, m/t]]}]]]; g[k0, n-k0, Length[l], 0]];
A[n_, k_] := If[k == 0, n^(2*n), Sum[b[n, j, k]*StirlingS2[n, j]* Binomial[n, j]*j!, {j, 0, n}]]; A[0, ] = 1; A[1, ] = 1;
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jan 27 2015, after Alois P. Heinz *)

A239761 Number of pairs of functions (f, g) on a set of n elements into itself satisfying f(g(x)) = f(x).

Original entry on oeis.org

1, 1, 10, 159, 3496, 98345, 3373056, 136535455, 6371523712, 336784920849, 19888195110400, 1297716672601151, 92721494240225280, 7199830049013964921, 603715489091812335616, 54366622743565012989375, 5233114241479255004839936, 536180296483497244155041825

Offset: 0

Chad Brewbaker, Mar 26 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..345

Cf. A181162, A239769, A239773.
Column k=1 of A245910.
Cf. A001622, A295188.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-j, i-1)*binomial(n, j)*j^j, j=0..n)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 17 2014

f4[n_] := Sum[n^k Sum[Binomial[n - 1, j]*n^(n - 1 - j)*StirlingS1[j + 1, k] *(-1)^(j + k + 1), {j, 0, n - 1}], {k, 1, n}] (* David Einstein, Oct 31 2016 *)

a(n) ~ 5^(-1/4) * ((1+sqrt(5))/2)^(3*n-1/2) * n^n / exp(2*n/(1+sqrt(5))). - Vaclav Kotesovec, Aug 07 2014
a(n) = Sum_{k = 1..n} A060281(n,k) n^k. - David Einstein, Oct 31 2016
a(n) = n! * [x^n] 1/(1 + LambertW(-x))^n. - Ilya Gutkovskiy, Oct 03 2017

a(6)-a(7) from Giovanni Resta, Mar 28 2014

a(8)-a(17) from Alois P. Heinz, Jul 17 2014

A245911 Number of pairs of endofunctions f, g on [n] satisfying f(g^n(i)) = f(i) for all i in [n].

Original entry on oeis.org

1, 1, 12, 207, 9184, 173225, 46097856, 729481375, 454190410752, 30607186160529, 12762075858688000, 1036636706945881151, 3080713389889966460928, 145084860487902521548921, 124325137916420574135066624, 56537825009822523196823829375

Offset: 0

Alois P. Heinz, Aug 06 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100

Main diagonal of A245910.

Cf. A245988.

with(combinat):
T:= proc(n, j) T(n, j):= binomial(n-1, j-1)*n^(n-j) end:
b:= proc(n, i, k) option remember; `if`(n=0 or i=1, x^n,
      expand(add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
      x^(igcd(i, k)*j)*b(n-i*j, i-1, k), j=0..n/i)))
    end:
a:= n-> add((p-> add(n^i*T(n, j)* coeff(p, x, i),
        i=0..degree(p)))(b(j$2, n)), j=0..n):
seq(a(n), n=0..20);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, k_] := b[n, i, k] = Function[{x}, If[n == 0 || i == 1, x^n, Expand[Sum[(i - 1)!^j*multinomial[n, Join[{n - i*j}, Array[i&, j]]]/j!* x^(GCD[i, k]*j)*b[n - i*j, i - 1, k][x], {j, 0, n/i}]]]];
a[n_] := If[n == 0, 1, Sum[Binomial[n - 1, j - 1]*n^(n - j)*b[j, j, n][n], {j, 0, n}]];
a /@ Range[0, 20] (* Jean-François Alcover, Oct 03 2019, after Alois P. Heinz *)

A239777 Number of pairs of functions f, g on a size n set into itself satisfying f(g(g(x))) = f(x).

Original entry on oeis.org

1, 1, 12, 249, 7744, 326745, 17773056, 1197261289, 97165842432, 9294416254161, 1030298497753600, 130527793649586201, 18685034341191917568, 2993332161753700720681, 532270629438646194561024, 104316725427708352041239625, 22394627939996943667912769536

Offset: 0

Chad Brewbaker, Mar 26 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A181162, A239769, A239773.

Column k=2 of A245910.

s:= proc(n, i) option remember; `if`(i=0, [[]],
       map(x-> seq([j, x[]], j=1..n), s(n, i-1)))
    end:
a:= proc(n) (l-> add(add(`if`([true$n]=[seq(evalb(
       f[g[g[i]]]=f[i]), i=1..n)], 1, 0), g=l), f=l))(s(n$2))
    end:
seq(a(n), n=0..5);  # Alois P. Heinz, Jul 16 2014
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
      expand(add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!
       *x^((2-irem(i, 2))*j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> add((p-> add(n^i*binomial(n-1, k-1)*n^(n-k)*
    coeff(p, x, i), i=0..degree(p)))(b(k$2)), k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 06 2014

c[n_] := c[n] =
    Sum[(n - 1)! n^(n - k)/(n - k)! t^(1 + Mod[k + 1, 2]), {k, 1, n}]
d[0] = 1
d[n_] := d[n] = Sum[Binomial[n - 1, k]*d[k]*c[n - k], {k, 0, n - 1}]
a[n_] := d[n] /. t -> n
Table[a[n], {n, 1, 10}] (* David Einstein, Nov 02 2016*)

a(6)-a(7) from Giovanni Resta, Mar 28 2014

a(8)-a(16) from Alois P. Heinz, Aug 06 2014

A245912 Number of pairs of endofunctions f, g on [n] satisfying f(g^3(i)) = f(i) for all i in [n].

Original entry on oeis.org

1, 1, 10, 207, 6856, 302345, 17812656, 1384059775, 131612023936, 14986421437329, 2051598980742400, 327546779949753551, 59790068922261980160, 12505503377433451819993, 2956768061598853524176896, 778675046844529953944661375, 228393818322135051214683406336

Offset: 0

Alois P. Heinz, Aug 06 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..150

Column k=3 of A245910.

with(combinat):
T:= proc(n, j) option remember; binomial(n-1, j-1)*n^(n-j) end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
      expand(add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
      x^(igcd(i, 3)*j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> add((p-> add(n^i*T(n, j)* coeff(p, x, i),
        i=0..degree(p)))(b(j$2)), j=0..n):
seq(a(n), n=0..20);

A245913 Number of pairs of endofunctions f, g on [n] satisfying f(g^4(i)) = f(i) for all i in [n].

Original entry on oeis.org

1, 1, 12, 249, 9184, 488745, 35463456, 3212948809, 369653885952, 52089606360081, 8863922597593600, 1752215974571247801, 402913941534323970048, 106177876504463493003001, 31939024924944619647750144, 10756222724503803551432639625, 4050020577581980281160989147136

Offset: 0

Alois P. Heinz, Aug 06 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..150

Column k=4 of A245910.

with(combinat):
T:= proc(n, j) option remember; binomial(n-1, j-1)*n^(n-j) end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
      expand(add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
      x^(igcd(i, 4)*j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> add((p-> add(n^i*T(n, j)* coeff(p, x, i),
        i=0..degree(p)))(b(j$2)), j=0..n):
seq(a(n), n=0..20);

A245914 Number of pairs of endofunctions f, g on [n] satisfying f(g^5(i)) = f(i) for all i in [n].

Original entry on oeis.org

1, 1, 10, 159, 3496, 173225, 15680736, 1618295455, 169456569472, 17962132149009, 2673715009888000, 652752279443748671, 185425990150444922880, 50400836024570702513401, 12815973354809222836596736, 3862679850655546273674429375, 1722827488179450551983866413056

Offset: 0

Alois P. Heinz, Aug 06 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..150

Column k=5 of A245910.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
      expand(add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
      x^(igcd(i, 5)*j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> add(binomial(n-1, j-1)*n^(n-j)*subs(x=n, b(j$2)), j=0..n):
seq(a(n), n=0..20);

A245915 Number of pairs of endofunctions f, g on [n] satisfying f(g^6(i)) = f(i) for all i in [n].

Original entry on oeis.org

1, 1, 12, 297, 11104, 578745, 46097856, 4892935369, 649893820416, 108501530261841, 21500188932505600, 4911081373878751401, 1343217062946070130688, 427404076347462876047113, 154711214699181287350984704, 64222714159878924347124911625

Offset: 0

Alois P. Heinz, Aug 06 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..150

Column k=6 of A245910.

with(combinat):
b:= proc(n, i) option remember; unapply(`if`(n=0 or i=1, x^n,
      expand(add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
      x^(igcd(i, 6)*j)*b(n-i*j, i-1)(x), j=0..n/i))), x)
    end:
a:= n-> add(binomial(n-1, j-1)*n^(n-j)*b(j$2)(n), j=0..n):
seq(a(n), n=0..20);

A245916 Number of pairs of endofunctions f, g on [n] satisfying f(g^7(i)) = f(i) for all i in [n].

Original entry on oeis.org

1, 1, 10, 159, 3496, 98345, 3373056, 729481375, 187564765312, 37157158911249, 6404841810150400, 1033733019005497151, 162392131536566261760, 25373998461297751027321, 13879265226159974639036416, 11935104515280353051806269375, 7611603822558997773619173031936

Offset: 0

Alois P. Heinz, Aug 06 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..150

Column k=7 of A245910.

with(combinat):
b:= proc(n, i) option remember; unapply(`if`(n=0 or i=1, x^n,
      expand(add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
      x^(igcd(i, 7)*j)*b(n-i*j, i-1)(x), j=0..n/i))), x)
    end:
a:= n-> add(binomial(n-1, j-1)*n^(n-j)*b(j$2)(n), j=0..n):
seq(a(n), n=0..20);

cp's OEIS Frontend

A062206 a(n) = n^(2n).

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A245980 Number A(n,k) of pairs of endofunctions f, g on [n] satisfying g^k(f(i)) = f(i) for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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Maple

Mathematica

A239761 Number of pairs of functions (f, g) on a set of n elements into itself satisfying f(g(x)) = f(x).

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A245911 Number of pairs of endofunctions f, g on [n] satisfying f(g^n(i)) = f(i) for all i in [n].

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Maple

Mathematica

A239777 Number of pairs of functions f, g on a size n set into itself satisfying f(g(g(x))) = f(x).

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Maple

Mathematica

Extensions

A245912 Number of pairs of endofunctions f, g on [n] satisfying f(g^3(i)) = f(i) for all i in [n].

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A245913 Number of pairs of endofunctions f, g on [n] satisfying f(g^4(i)) = f(i) for all i in [n].

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Maple

A245914 Number of pairs of endofunctions f, g on [n] satisfying f(g^5(i)) = f(i) for all i in [n].

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Maple

A245915 Number of pairs of endofunctions f, g on [n] satisfying f(g^6(i)) = f(i) for all i in [n].

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