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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A242054 Column 3 of square array A246072 / n!.

Original entry on oeis.org

1, 1, 5, 18, 75, 396, 2052, 11586, 71787, 458352, 3103668, 22202874, 164999826, 1281692088, 10371684312, 86973240204, 755908929603, 6794220017664, 63008287321788, 602270212069098, 5924679849081126, 59897824980579576, 621672797654084520, 6616610400436719588
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 13 2014

Keywords

Links

Crossrefs

Cf. A246072.

Programs

  • Maple
    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, 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, (2*n)!, b(2*n, n, k)):
seq(A(n,3)/n!, n = 0..20); # after Alois P. Heinz
  • Mathematica
    multinomial[n_, k_List] := n!/Times @@ (k!);
M = multinomial;
b[n_, k_, p_] := b[n, k, p] = Module[{l, g}, l = Sort[Divisors[p]]; g[k0_, m_, i_, t_] := g[k0, m, i, t] = Module[{d}, d = l[[i]]; If[i == 1, m!, Sum[M[k0, Join[{k0-(d-t)j}, Table[d-t, {j}]]]/j! (d-1)!^j M[m, Join[{m - t j}, Table[t, {j}]]] If[d-t == 1, g[k0 - (d-t) j, m - t j, i-1, 0], g[k0 - (d-t)j, m - t j, i, t+1]], {j, 0, Min[k0/(d-t), If[t == 0, Infinity, m/t]]}]]]; g[k, n-k, Length[l], 0]];
A[n_, k_] := If[k == 0, (2n)!, b[2n, n, k]];
a[n_] := A[n, 3]/n!;
a /@ Range[0, 100] (* Jean-François Alcover, Nov 13 2020, after Alois P. Heinz in A246072 *)

Formula

Recurrence: n*(125*n^10 - 8575*n^9 + 249165*n^8 - 3972421*n^7 + 38651424*n^6 - 241441049*n^5 + 985299581*n^4 - 2598873155*n^3 + 4233949973*n^2 - 3823560792*n + 1433318628)*a(n) = (n-1)*(250*n^10 - 17275*n^9 + 505255*n^8 - 7939307*n^7 + 74253125*n^6 - 433278950*n^5 + 1592429495*n^4 - 3581735158*n^3 + 4503495303*n^2 - 2437269642*n + 30743664)*a(n-1) + (250*n^12 - 17025*n^11 + 485380*n^10 - 7526552*n^9 + 70321082*n^8 - 411180662*n^7 + 1481140960*n^6 - 2907087063*n^5 + 1110382294*n^4 + 8175223780*n^3 - 18648517428*n^2 + 16445473512*n - 5256191136)*a(n-2) + (125*n^13 - 7575*n^12 + 179440*n^11 - 1844901*n^10 + 1377531*n^9 + 175414789*n^8 - 2147090919*n^7 + 13587878007*n^6 - 53501005344*n^5 + 136256919050*n^4 - 221250046545*n^3 + 215137237254*n^2 - 107999226624*n + 18548116944)*a(n-3) - (3125*n^12 - 230275*n^11 + 7312925*n^10 - 129375074*n^9 + 1424467574*n^8 - 10349546810*n^7 + 51109969432*n^6 - 173096865441*n^5 + 398028650726*n^4 - 601201146808*n^3 + 556233509382*n^2 - 271925267076*n + 47484061632)*a(n-4) - (n-4)*(3125*n^12 - 222875*n^11 + 6789125*n^10 - 115816245*n^9 + 1238520315*n^8 - 8794888450*n^7 + 42652555528*n^6 - 142304505773*n^5 + 322756817266*n^4 - 480186892231*n^3 + 434864239701*n^2 - 204163716870*n + 31770252360)*a(n-5) - (n-5)*(n-4)*(1375*n^12 - 95825*n^11 + 2889440*n^10 - 49131951*n^9 + 525311115*n^8 - 3732302998*n^7 + 18093070721*n^6 - 60182645063*n^5 + 135414119977*n^4 - 198015832273*n^3 + 172828363290*n^2 - 74116518432*n + 7983006192)*a(n-6) - (n-6)*(n-5)*(n-4)*(250*n^10 - 13325*n^9 + 316180*n^8 - 4021155*n^7 + 29819852*n^6 - 134332700*n^5 + 368917385*n^4 - 590456906*n^3 + 472005585*n^2 - 82277310*n - 69925968)*a(n-7) + (n-7)*(n-6)*(n-5)*(n-4)*(250*n^10 - 14025*n^9 + 329680*n^8 - 4094477*n^7 + 29774855*n^6 - 132986602*n^5 + 367961264*n^4 - 609685740*n^3 + 541560093*n^2 - 175805898*n - 27276984)*a(n-8) - (n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(125*n^10 - 7325*n^9 + 177615*n^8 - 2272801*n^7 + 17127047*n^6 - 80049056*n^5 + 235218311*n^4 - 424843636*n^3 + 437741568*n^2 - 212268240*n + 23612904)*a(n-9).
a(n) ~ n^(2*n/3) * exp(-2*n/3 + n^(2/3) + 4/3*n^(1/3) - 8/9) / sqrt(3). - Vaclav Kotesovec, Aug 13 2014

A246070 Number A(n,k) of endofunctions f on [2n] satisfying f^k(i) = 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, 4, 1, 2, 256, 1, 3, 16, 46656, 1, 2, 50, 216, 16777216, 1, 3, 36, 1626, 4096, 10000000000, 1, 2, 56, 1440, 83736, 100000, 8916100448256, 1, 3, 16, 2688, 84624, 6026120, 2985984, 11112006825558016, 1, 2, 70, 720, 215760, 7675200, 571350096, 105413504, 18446744073709551616
Offset: 0

Views

Author

Alois P. Heinz, Aug 12 2014

Keywords

Examples

			Square array A(n,k) begins:
0 :            1,      1,       1,       1,        1,        1, ...
1 :            4,      2,       3,       2,        3,        2, ...
2 :          256,     16,      50,      36,       56,       16, ...
3 :        46656,    216,    1626,    1440,     2688,      720, ...
4 :     16777216,   4096,   83736,   84624,   215760,    94816, ...
5 :  10000000000, 100000, 6026120, 7675200, 24899120, 11218000, ...

Links

Crossrefs

Columns k=0-3 give: A085534, A062971, A245141, A245959.
Main diagonal gives A246071.
Cf. A246072 (the same for permutations).

Programs

  • Maple
    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, (2*n)^(2*n), b(2*n, n, k)):
seq(seq(A(n, d-n), n=0..d), d=0..10);
  • Mathematica
    multinomial[n_, k_List] := n!/Times @@ (k!); M = multinomial;
b[n_, k0_, p_] := Module[{l, g}, l = Divisors[p];
g[k_, m_, i_, t_] := g[k, m, i, t] = Module[{d, j}, d = l[[i]];
    If[i == 1, If[m == 0, 1, n^m], Sum[M[k, Join[{k - (d - t)*j},
    Table[d - t, {j}]]]/j!*If[j == 0, 1, (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, {}, m/t]]}]]];
g[k0, n - k0, Length[l], 0]];
A[n_, k_] := If[k == 0, If[n == 0, 1, (2n)^(2n)], b[2*n, n, k]];
Table[A[n, d - n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 27 2016, after Alois P. Heinz, updated Jan 01 2021 *)

A246073 Number of permutations p on [2n] satisfying p^n(i) = i for all i in [n].

Original entry on oeis.org

1, 1, 10, 108, 6672, 109200, 45007920, 983324160, 665546434560, 60174422501760, 32648180513760000, 4656975300322329600, 13859947861644771532800, 1193599114668580293273600, 1257285172911535450293811200, 766119340152013216053484800000
Offset: 0

Views

Author

Alois P. Heinz, Aug 12 2014

Keywords

Comments

Conjecture: Lim inf n->infinity a(n) / (((n-1)!)^2 * 4^(n-1) / sqrt(n)) = 1.128... . - Vaclav Kotesovec, Aug 14 2014

Examples

			a(2) = 10: (1,2,3,4), (1,2,4,3), (1,3,2,4), (1,4,3,2), (2,1,3,4), (2,1,4,3), (3,2,1,4), (3,4,1,2), (4,2,3,1), (4,3,2,1).
a(3) = 108: (1,2,3,4,5,6), (1,2,3,4,6,5), (1,2,3,5,4,6), ... (6,4,2,3,1,5), (6,5,1,2,4,3), (6,5,2,1,3,4).

Links

Crossrefs

Main diagonal of A246072.

Programs

  • Maple
    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, 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-> `if`(n=0, 1, b(2*n, n, n)):
seq(a(n), n=0..20);
  • Mathematica
    multinomial[n_, k_List] := n!/Times @@ (k!); M = multinomial;
b[n_, k_, p_] := b[n, k, p] = Module[{l, g}, l = Sort[Divisors[p]]; g[k0_, m_, i_, t_] := g[k0, m, i, t] = Module[{d}, d = l[[i]]; If[i == 1, m!, Sum[M[k0, Join[{k0 - (d - t) j}, Table[d - t, {j}]]]/j! (d - 1)!^j M[m, Join[{m - t j}, Table[t, {j}]]] If[d - t == 1, g[k0 - (d - t) j, m - t j, i - 1, 0], g[k0 - (d - t) j, m - t j, i, t + 1]], {j, 0, Min[k0/(d - t), If[t == 0, Infinity, m/t]]}]]]; g[k, n - k, Length[l], 0]];
a[n_] := If[n == 0, 1, b[2n, n, n]];
a /@ Range[0, 20] (* Jean-François Alcover, Nov 13 2020, after Alois P. Heinz *)

Formula

a(n) = A246072(2n,n).
Showing 1-3 of 3 results.

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