A245980 -id:A245980 - 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

A245910 Number A(n,k) of pairs of endofunctions f, g on [n] satisfying f(g^k(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, 10, 729, 1, 1, 12, 159, 65536, 1, 1, 10, 249, 3496, 9765625, 1, 1, 12, 207, 7744, 98345, 2176782336, 1, 1, 10, 249, 6856, 326745, 3373056, 678223072849, 1, 1, 12, 159, 9184, 302345, 17773056, 136535455, 281474976710656

Offset: 0

Alois P. Heinz, Aug 06 2014

			Square array A(n,k) begins:
0 :        1,     1,      1,      1,      1,      1, ...
1 :        1,     1,      1,      1,      1,      1, ...
2 :       16,    10,     12,     10,     12,     10, ...
3 :      729,   159,    249,    207,    249,    159, ...
4 :    65536,  3496,   7744,   6856,   9184,   3496, ...
5 :  9765625, 98345, 326745, 302345, 488745, 173225, ...

Alois P. Heinz, Antidiagonals n = 0..100, flattened

Columns k=0-10 give: A062206, A239761, A239777, A245912, A245913, A245914, A245915, A245916, A245917, A245918, A245919.
Main diagonal gives A245911.
Cf. A061356, A245980.

with(combinat):
b:= proc(n, i, k) 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, k)*j)*b(n-i*j, i-1, k)(x), j=0..n/i))), x)
    end:
A:= (n, k)-> `if`(k=0, n^(2*n), add(binomial(n-1, j-1)*n^(n-j)*
              b(j$2, k)(n), j=0..n)):
seq(seq(A(n, d-n), n=0..d), d=0..10);

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[0, ] = 1; A[n, k_] := If[k == 0, n^(2n), Sum[Binomial[n-1, j-1]*n^(n-j)* b[j, j, k][n], {j, 0, n}]]; Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 04 2015, after Alois P. Heinz *)

A241015 Number of pairs of endofunctions f, g on [n] satisfying g(g(g(f(i)))) = f(i) for all i in [n].

Original entry on oeis.org

1, 1, 6, 141, 6184, 387545, 33404256, 3891981205, 592320594048, 113184611671473, 26327424526220800, 7302855260707822541, 2381136881374877847552, 901709366369630531857417, 392234247731566637785780224, 194028806625479344354551301125

Offset: 0

Alois P. Heinz, Aug 07 2014

nonn

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

Cf. A048993, A239750, A239771, A245958.

Column k=3 of A245980.

with(combinat): M:=multinomial:
b:= proc(n, k) local l, g; l, g:= [1, 3],
      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-> add(b(n, j)*stirling2(n, j)*binomial(n, j)*j!, j=0..n):
seq(a(n), n=0..20);

multinomial[n_, k_] := n!/Times @@ (k!); M = multinomial; b[n_, k0_] := Module[{l, g}, l = {1, 3}; 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]]]*g[k-(d-t)*j, m-t*j, Sequence @@ If[d-t==1, {i-1, 0}, {i, t+1}]], {j, 0, Min[k/(d-t), If[t==0, Infinity, m/t]]}]]]; g[k0, n-k0, Length[l], 0]]; a[0] = 1; a[n_] := Sum[b[n, j]*StirlingS2[n, j]*Binomial[n, j]*j!, {j, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 13 2017, translated from Maple *)

a(n) = Sum_{k=0..n} C(n,k) * A048993(n,k) * k! * A245958(n,k).

A239771 Number of pairs of functions (f,g) from a size n set into itself satisfying f(x) = g(g(f(x))).

Original entry on oeis.org

1, 1, 10, 213, 8056, 465945, 37823616, 4075467781, 560230714240, 95369455852497, 19643693349548800, 4805295720474420501, 1374890520609054683136, 454286686896040037996905, 171479277693049020232695808, 73262491601904459123264721125, 35143072854722729593790081499136

Offset: 0

Chad Brewbaker, Mar 26 2014

nonn

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

Cf. A000085, A048993, A181162, A239769, A239773, A245348, A241015.

Column k=2 of A245980.

g:= proc(n) g(n):= `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
a:= n-> add(binomial(n, k)*Stirling2(n, k)*k!*
        add(binomial(n-k, i)*binomial(k, i)*i!*
        g(k-i)*n^(n-k-i), i=0..min(k, n-k)), k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 18 2014

g[n_] := g[n] = If[n < 2, 1, g[n-1] + (n-1)*g[n-2]];
a[n_] := If[n == 0, 1, Sum[Binomial[n, k]*StirlingS2[n, k]*k!*Sum[ Binomial[n-k, i]*Binomial[k, i]*i!*g[k-i]*n^(n-k-i), {i, 0, Min[k, n-k]} ], {k, 0, n}]];
a /@ Range[0, 20] (* Jean-François Alcover, Oct 03 2019, after Alois P. Heinz *)

a(n) = Sum_{k=0..n} C(n,k) * A048993(n,k) * k! * A245348(n,k). - Alois P. Heinz, Jul 18 2014

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

a(8)-a(16) from Alois P. Heinz, Jul 18 2014

A239750 Number of ordered pairs of endofunctions (f,g) on a set of n elements satisfying g(f(x)) = f(f(f(x))).

Original entry on oeis.org

1, 1, 6, 87, 2200, 84245, 4492656, 315937195, 28186856832, 3099006365769, 410478164588800, 64323095036300111, 11748771067445148672, 2470422069374379054493, 591735532838657160296448, 160004357420756572368889875, 48458574881000820765562863616

Offset: 0

Chad Brewbaker, Mar 26 2014

nonn

As observed by Yuval Filmus, this also counts pairs (f,g) that satisfy g(f(x)) = f^{k}(x) for k >= 1. - Chad Brewbaker, Mar 27 2014

Alois P. Heinz, Table of n, a(n) for n = 0..200
Yuval Filmus, Answer to 'A curious Wilf equivalence class of function compositions', Mar 27 2014

Cf. A000248, A000949, A048993, A181162, A241015.

Column k=1 of A245980.

a:= n-> add(binomial(n, k)*k^n*(n-1)^(n-k), k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 23 2014

a[n_] := If[n<2, 1, Sum[Binomial[n, k]*k^n*(n-1)^(n-k), {k, 0, n}]];
a /@ Range[0, 20] (* Jean-François Alcover, Oct 03 2019, after Alois P. Heinz *)

a(n) = Sum_{k=0..n} C(n,k) * k^n * (n-1)^(n-k) = Sum_{k=0..n} C(n,k) * A048993(n,k) * k! * n^(n-k). - Alois P. Heinz, Jul 23 2014

a(6)-a(7) from Giovanni Resta, Mar 26 2014

a(8)-a(16) from Alois P. Heinz, Jul 17 2014

A245988 Number of pairs of endofunctions f, g on [n] satisfying g^n(f(i)) = f(i) for all i in [n].

Original entry on oeis.org

1, 1, 10, 141, 9592, 159245, 86252976, 908888155, 1682479423360, 128805405787953, 93998774487116800, 1099662085349496911, 44830846497021739693056, 147548082727234113659293, 3534565745374740945151080448, 1613371163531618738559582856125

Offset: 0

Alois P. Heinz, Aug 08 2014

nonn

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

Main diagonal of A245980.

Cf. A245911.

with(numtheory): with(combinat): M:=multinomial:
a:= proc(n) option remember; local l, g; l, g:= sort([divisors(n)[]]),
      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; forget(g);
      `if`(n=0, 1, add(g(j, n-j, nops(l), 0)*
      stirling2(n, j)*binomial(n, j)*j!, j=0..n))
    end:
seq(a(n), n=0..20);

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, ] = A[1, ] = 1;
a[n_] := A[n, n];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 29 2022, after Alois P. Heinz in A245980 *)

a(n) = A245980(n,n).

A245981 Number of pairs of endofunctions f, g on [n] satisfying g^4(f(i)) = f(i) for all i in [n].

Original entry on oeis.org

1, 1, 10, 213, 9592, 682545, 69119136, 9284636221, 1597922254720, 344058384011553, 90769698354764800, 28762381447366581861, 10751918763610399942656, 4671451080680229243978385, 2331208959412708894563057664, 1323549917511104579568688414125

Offset: 0

Alois P. Heinz, Aug 08 2014

nonn

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

Column k=4 of A245980.

with(combinat): M:=multinomial:
b:= proc(n, k) local l, g; l, g:= [1, 2, 4],
      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-> add(b(n, j)*stirling2(n, j)*binomial(n, j)*j!, j=0..n):
seq(a(n), n=0..20);

A245982 Number of pairs of endofunctions f, g on [n] satisfying g^5(f(i)) = f(i) for all i in [n].

Original entry on oeis.org

1, 1, 6, 87, 2200, 159245, 22460976, 3841485235, 725338311552, 150719206127769, 35342379764876800, 9829163373723941951, 3429714088052022223872, 1523614487096970692512933, 823050850772773045911871488, 507838824721407879972472444875

Offset: 0

Alois P. Heinz, Aug 08 2014

nonn

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

Column k=5 of A245980.

with(combinat): M:=multinomial:
b:= proc(n, k) local l, g; l, g:= [1, 5],
      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-> add(b(n, j)*stirling2(n, j)*binomial(n, j)*j!, j=0..n):
seq(a(n), n=0..20);

A245983 Number of pairs of endofunctions f, g on [n] satisfying g^6(f(i)) = f(i) for all i in [n].

Original entry on oeis.org

1, 1, 10, 267, 12040, 826245, 86252976, 12661148311, 2428606888576, 585229569018921, 172640322717932800, 60933514918456147011, 25283156000087876668416, 12189356237264450125373869, 6769905753950075837079906304, 4297777320612236566890778059375

Offset: 0

Alois P. Heinz, Aug 08 2014

nonn

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

Column k=6 of A245980.

with(combinat): M:=multinomial:
b:= proc(n, k) local l, g; l, g:= [1, 2, 3, 6],
      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->add(b(n, j)*stirling2(n, j)*binomial(n, j)*j!, j=0..n):
seq(a(n), n=0..20);

A245984 Number of pairs of endofunctions f, g on [n] satisfying g^7(f(i)) = f(i) for all i in [n].

Original entry on oeis.org

1, 1, 6, 87, 2200, 84245, 4492656, 908888155, 357260391552, 135745499491209, 49743738690284800, 18418196210352315311, 7088670872640238205952, 2879857079508362958098653, 1254944121383140772128247808, 610054332530467361553695923875

Offset: 0

Alois P. Heinz, Aug 08 2014

nonn

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

Column k=7 of A245980.

with(combinat): M:=multinomial:
b:= proc(n, k) local l, g; l, g:= [1, 7],
      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-> add(b(n, j)*stirling2(n, j)*binomial(n, j)*j!, j=0..n):
seq(a(n), n=0..20);

cp's OEIS Frontend

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

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A245910 Number A(n,k) of pairs of endofunctions f, g on [n] satisfying f(g^k(i)) = f(i) for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A241015 Number of pairs of endofunctions f, g on [n] satisfying g(g(g(f(i)))) = f(i) for all i in [n].

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A239771 Number of pairs of functions (f,g) from a size n set into itself satisfying f(x) = g(g(f(x))).

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A239750 Number of ordered pairs of endofunctions (f,g) on a set of n elements satisfying g(f(x)) = f(f(f(x))).

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A245988 Number of pairs of endofunctions f, g on [n] satisfying g^n(f(i)) = f(i) for all i in [n].

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A245981 Number of pairs of endofunctions f, g on [n] satisfying g^4(f(i)) = f(i) for all i in [n].

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A245982 Number of pairs of endofunctions f, g on [n] satisfying g^5(f(i)) = f(i) for all i in [n].

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