cp's OEIS Frontend

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 1-5 of 5 results.

A350078 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose second-largest component has size exactly k; n >= 0, 0 <= k <= floor(n/2).

Original entry on oeis.org

1, 1, 3, 1, 17, 10, 142, 87, 27, 1569, 911, 645, 21576, 11930, 10260, 2890, 355081, 189610, 174132, 104720, 6805296, 3543617, 3229275, 2493288, 705740, 148869153, 76060087, 67843521, 60223520, 34424208, 3660215680, 1842497914, 1605373560, 1530575960, 1051155000, 310181886
Offset: 0

Views

Author

Steven Finch, Dec 12 2021

Keywords

Comments

An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.
If the mapping has no second component, then its second-largest component is defined to have size 0.

Examples

			Triangle begins:
       1;
       1;
       3,      1;
      17,     10;
     142,     87,     27;
    1569,    911,    645;
   21576,  11930,  10260,   2890;
  355081, 189610, 174132, 104720;
  ...

Links

Crossrefs

Column 0 gives gives 1 together with A001865.
Row sums give A000312.
Cf. A001865, A350079, A350080, A350081, A350275, A350276.

Programs

  • Maple
    g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, l) option remember; `if`(n=0, x^l[1], add(g(i)*
      b(n-i, sort([l[], i])[-2..-1])*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$2])):
seq(T(n), n=0..10);  # Alois P. Heinz, Dec 17 2021
  • Mathematica
    g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, l_] := g[n, l] = If[n == 0, x^l[[1]], Sum[g[i]*b[n - i, Sort[ Append[l, i]][[-2 ;; -1]]]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := With[{p = b[n, {0, 0}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

Extensions

More terms (three rows) from Alois P. Heinz, Dec 15 2021

A350079 Triangle read by rows: T(n,k) is the number of endofunctions on [n] whose second-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-1).

Original entry on oeis.org

1, 1, 3, 1, 17, 1, 9, 142, 19, 27, 68, 1569, 201, 135, 510, 710, 21576, 2921, 3465, 2890, 6390, 9414, 355081, 50233, 63630, 20230, 84490, 98847, 151032, 6805296, 1004599, 1196181, 918680, 705740, 1493688, 1812384, 2840648, 148869153, 22872097, 26904339, 23943752, 6351660, 28072548, 30810528, 38348748, 61247664
Offset: 0

Views

Author

Steven Finch, Dec 12 2021

Keywords

Comments

An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.
If the mapping has no second component, then its second-smallest component is defined to have size 0.

Examples

			Triangle begins:
       1;
       1;
       3,     1;
      17,     1,     9;
     142,    19,    27,    68;
    1569,   201,   135,   510,   710;
   21576,  2921,  3465,  2890,  6390,  9414;
  355081, 50233, 63630, 20230, 84490, 98847, 151032;
  ...

Links

Crossrefs

Column 0 gives gives 1 together with A001865.
Row sums give A000312.
Cf. A001865, A350078, A350080, A350081, A350275, A350276

Programs

  • Maple
    g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, l) option remember; `if`(n=0, x^subs(infinity=0, l)[2],
      add(b(n-i, sort([l[], i])[1..2])*g(i)*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [infinity$2])):
seq(T(n), n=0..12);  # Alois P. Heinz, Dec 17 2021
  • Mathematica
    g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, l_] := b[n, l] = If[n == 0, x^(l /. Infinity -> 0)[[2]], Sum[b[n - i, Sort[Append[l, i]][[1;;2]]]*g[i]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := With[{p = b[n, {Infinity, Infinity}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

Extensions

More terms (two rows) from Alois P. Heinz, Dec 15 2021

A350080 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose third-largest component has size exactly k; n >= 0, 0 <= k <= floor(n/3).

Original entry on oeis.org

1, 1, 4, 26, 1, 237, 19, 2789, 336, 40270, 5981, 405, 689450, 115193, 18900, 13657756, 2459955, 659505, 307348641, 58366045, 20330163, 1375640, 7745565616, 1530739594, 623758590, 99936200, 216114310994, 44076571672, 19795671225, 5325116720
Offset: 0

Views

Author

Steven Finch, Dec 12 2021

Keywords

Comments

An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.
If the mapping has no third component, then its third-largest component is defined to have size 0.

Examples

			Triangle begins:
       1;
       1;
       4;
      26,     1;
     237,    19;
    2789,   336;
   40270,   5981,   405;
  689450, 115193, 18900;
  ...

Links

Crossrefs

Row sums give A000312.
Cf. A001865, A350078, A350079, A350081, A350275, A350276.

Programs

  • Maple
    g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, l) option remember; `if`(n=0, x^l[1], add(g(i)*
      b(n-i, sort([l[], i])[-3..-1])*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$3])):
seq(T(n), n=0..12);  # Alois P. Heinz, Dec 17 2021
  • Mathematica
    g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, l_] := b[n, l] = If[n == 0, x^l[[1]], Sum[g[i]*b[n - i, Sort[ Append[l, i]][[-3 ;; -1]]]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := With[{p = b[n, {0, 0, 0}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

Extensions

More terms (4 rows) from Alois P. Heinz, Dec 16 2021

A350081 Triangle read by rows: T(n,k) is the number of endofunctions on [n] whose third-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-2).

Original entry on oeis.org

1, 1, 4, 26, 1, 237, 1, 18, 2789, 31, 135, 170, 40270, 386, 810, 3060, 2130, 689450, 6574, 13545, 36295, 44730, 32949, 13657756, 129291, 327285, 323680, 944300, 790776, 604128, 307348641, 2910709, 7207137, 6602120, 15476580, 18780930, 16311456, 12782916
Offset: 0

Views

Author

Steven Finch, Dec 12 2021

Keywords

Comments

An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.
If the mapping has no third component, then its third-smallest component is defined to have size 0.

Examples

			Triangle begins:
       1;
       1;
       4;
      26,    1;
     237,    1,    18;
    2789,   31,   135,   170;
   40270,  386,   810,  3060,  2130;
  689450, 6574, 13545, 36295, 44730, 32949;
  ...

Links

Crossrefs

Row sums give A000312.
Cf. A001865, A350078, A350079, A350080, A350275, A350276.

Programs

  • Maple
    g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, l) option remember; `if`(n=0, x^subs(infinity=0, l)[3],
      add(b(n-i, sort([l[], i])[1..3])*g(i)*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [infinity$3])):
seq(T(n), n=0..12);  # Alois P. Heinz, Dec 17 2021
  • Mathematica
    g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, l_] := b[n, l] = If[n == 0, x^(l /. Infinity -> 0)[[3]], Sum[b[n - i, Sort[Append[l, i]][[1 ;; 3]]]*g[i]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := With[{p = b[n, {Infinity, Infinity, Infinity}]}, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

Extensions

More terms (two rows) from Alois P. Heinz, Dec 16 2021

A350275 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose fourth-largest component has size exactly k; n >= 0, 0 <= k <= floor(n/4).

Original entry on oeis.org

1, 1, 4, 27, 255, 1, 3094, 31, 45865, 791, 803424, 20119, 16239720, 528991, 8505, 372076163, 14689441, 654885, 9529560676, 435580164, 34859160, 269819334245, 13846282341, 1646054025, 8369112382488, 471890017358, 73811825010, 1286223400
Offset: 0

Views

Author

Steven Finch, Dec 22 2021

Keywords

Comments

An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.
If the mapping has no fourth component, then its fourth-largest component is defined to have size 0.

Examples

			Triangle begins:
       1;
       1;
       4;
      27;
     255,     1;
    3094,    31;
   45865,   791;
  803424, 20119;
  ...

Links

Crossrefs

Row sums give A000312.
Cf. A001865, A350078, A350079, A350080, A350081, A350276.

Programs

  • Maple
    g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, l) option remember; `if`(n=0, x^l[1], add(g(i)*
      b(n-i, sort([l[], i])[-4..-1])*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$4])):
seq(T(n), n=0..14);  # Alois P. Heinz, Dec 22 2021
  • Mathematica
    g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, l_] := b[n, l] = If[n == 0, x^l[[1]], Sum[g[i]*b[n - i, Sort[ Append[l, i]][[-4 ;; -1]]]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := With[{p = b[n, {0, 0, 0, 0}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)
Showing 1-5 of 5 results.

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