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.

A135302 Square array of numbers A(n,k) (n>=0, k>=0) of transitive reflexive early confluent binary relations R on n labeled elements where |{y : xRy}| <= k for all x, read by antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 4, 1, 1, 0, 1, 13, 4, 1, 1, 0, 1, 62, 26, 4, 1, 1, 0, 1, 311, 168, 26, 4, 1, 1, 0, 1, 1822, 1416, 243, 26, 4, 1, 1, 0, 1, 11593, 13897, 2451, 243, 26, 4, 1, 1, 0, 1, 80964, 153126, 29922, 2992, 243, 26, 4, 1, 1, 0, 1, 608833, 1893180, 420841, 41223, 2992, 243, 26, 4, 1, 1
Offset: 0

Views

Author

Alois P. Heinz, Dec 04 2007

Keywords

Comments

R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.

Examples

			Table A(n,k) begins:
  1, 1,   1,    1,    1,    1, ...
  0, 1,   1,    1,    1,    1, ...
  0, 1,   4,    4,    4,    4, ...
  0, 1,  13,   26,   26,   26, ...
  0, 1,  62,  168,  243,  243, ...
  0, 1, 311, 1416, 2451, 2992, ...

References

  • A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.

Links

Crossrefs

Columns k=0-10 give: A000007, A000012, A135312, A210911, A210912, A210913, A210914, A210915, A210916, A210917, A210918.
Main diagonal gives A052880.
A(n,n)-A(n,n-1) gives A000670.
Cf. A135313.

Programs

  • Maple
    t:= proc(k) option remember; `if`(k<0, 0,
       unapply(exp(add(x^m/m! *t(k-m)(x), m=1..k)), x))
    end:
A:= proc(n, k) option remember;
      coeff(series(t(k)(x), x, n+1), x, n) *n!
    end:
seq(seq(A(d-i, i), i=0..d), d=0..15);
  • Mathematica
    t[0, ] = 1; t[k, x_] := t[k, x] = Exp[Sum[x^m/m!*t[k-m, x], {m, 1, k}]]; a[0, 0] = 1; a[, 0] = 0; a[n, k_] := SeriesCoefficient[t[k, x], {x, 0, n}]*n!; Table[a[n-k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 06 2013, after Maple *)

Formula

E.g.f. of column k=0: t_0(x) = 1; e.g.f. of column k>0: t_k(x) = exp (Sum_{m=1..k} x^m/m! * t_{k-m}(x)).
A(n,k) = Sum_{i=0..k} A135313(n,i).

A135429 Number of transitive reflexive binary relations R on n labeled elements where |{y : xRy}| <= 3 for all x.

Original entry on oeis.org

1, 1, 4, 29, 210, 2116, 25522, 362832, 6000276, 113593688, 2434603356, 58523364604, 1565365441708, 46273309903536, 1502773485741816, 53336787604185656, 2059209704215556448, 86117458019804680576, 3886421648246467359364, 188615552477984650605744
Offset: 0

Views

Author

Alois P. Heinz, Dec 12 2007

Keywords

References

  • A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.

Links

Crossrefs

Cf. A135312, A025035, A006882, A008277, A000142.

Programs

  • Maple
    A025035:= proc(n) option remember; (3*n)! /n! /(6^n) end:
z:= proc(n) option remember; add(binomial(n,k+k) *doublefactorial(k+k-1) *k^(n-k-k), k=0..floor(n/2)) end:
r:= proc(n) option remember; n! * add(add(add(add(Stirling2(e,d) *a^(d+i) *(a*(a+1)/2)^(n-i-i-e-d-a) /a! /(n-i-i-e-d-a)! /i! /e! /(2^i), a=0..(n-i-i-e-d)), d=0..min(e,n-i-i-e)), e=0..(n-i-i)), i=0..floor(n/2)) end:
a:= proc(n) option remember; n! *add(add(A025035(i) *z(j) *r(n-3*i-j) /(3*i)! /j! /(n-3*i-j)!, j=0..(n-3*i)), i=0..floor(n/3)) end:
seq(a(n), n=0..30);
  • Mathematica
    Unprotect[Power]; 0^0 = 1; A025035[n_] := A025035[n] = (3n)!/n!/6^n; z[n_] := z[n] = Sum[Binomial[n, k+k]*(k+k-1)!!*k^(n-k-k), {k, 0, Floor[n/2]}]; r[n_] := r[n] = n!*Sum[Sum[Sum[Sum[StirlingS2[e, d]*a^(d+i)*(a*(a+1)/2)^(n-i-i-e-d-a)/a!/(n-i-i-e-d-a)!/i!/e!/2^i, {a, 0, n-i-i-e-d}], {d, 0, Min[e, n-i-i-e]}], {e, 0, n-i-i}], {i, 0, Floor[n/2]}]; a[n_] := a[n] = n!*Sum[Sum[A025035[i]*z[j]*r[n-3*i-j]/(3i)!/j!/(n-3*i-j)!, {j, 0, n-3*i}], {i, 0, Floor[n/3]}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 05 2014, after Alois P. Heinz *)

Formula

a(n) = see program.

A135458 Number of transitive reflexive binary relations R on n labeled elements where max_{x}(|{y : xRy}|)=3.

Original entry on oeis.org

0, 0, 0, 16, 148, 1805, 23700, 351239, 5919312, 112984855, 2429692570, 58481205365, 1564981962720, 46269631044377, 1502736397861062, 53336395962363115, 2059205384354896000, 86117408372404734527, 3886421055028996900050, 188615545123265662965785
Offset: 0

Views

Author

Alois P. Heinz, Dec 15 2007

Keywords

Examples

			a(3)=16 because there are 16 relations of the given kind for 3 elements:
1R2, 2R1, 1R3, 3R1, 2R3, 3R2;
1R2, 1R3, 2R3, 3R2;
2R1, 2R3, 1R3, 3R1;
3R1, 3R2, 1R2, 2R1;
1R2, 2R1, 1R3, 2R3;
1R3, 3R1, 1R2, 3R2;
2R3, 3R2, 2R1, 3R1;
1R2, 2R3, 1R3;
1R3, 3R2, 1R2;
2R1, 1R3, 2R3;
2R3, 3R1, 2R1;
3R1, 1R2, 3R2;
3R2, 2R1, 3R1;
1R2, 1R3;
2R1, 2R3;
3R1, 3R2;
(the reflexive relationships 1R1, 2R2, 3R3 have been omitted for brevity)

References

  • A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.

Links

Crossrefs

Cf. A135429, A135312, A025035, A008277, A006882, A007318, A000248, A000142.

Programs

  • Maple
    A025035:= proc(n) option remember; (3*n)! /n! /(6^n); end:
z:= proc(n) option remember; add(binomial(n, k+k) *doublefactorial(k+k-1) *k^(n-k-k), k=0..floor(n/2)); end:
r:= proc(n) option remember; n! * add(add(add(add(Stirling2(e, d) *a^(d+i) *(a*(a+1)/2)^(n-i-i-e-d-a) /a! /(n-i-i-e-d-a)! /i! /e! /(2^i), a=0..(n-i-i-e-d)), d=0..min(e, n-i-i-e)), e=0..(n-i-i)), i=0..floor(n/2)) end:
A135429:= proc(n) option remember; n! *add(add(A025035(i) *z(j) *r(n-3*i-j) /(3*i)! /j! /(n-3*i-j)!, j=0..(n-3*i)), i=0..floor(n/3)) end:
A000248:= proc(n) add(binomial(n, i)*(n-i)^i, i=0..n) end:
A135312:= proc(n) option remember; add(binomial(n, i+i)*doublefactorial(i+i-1)*A000248(n-i-i), i=0..floor(n/2)) end:
a:= proc(n) A135429(n)-A135312(n) end:
seq(a(i), i=0..30);
  • Mathematica
    Unprotect[Power]; 0^0 = 1; A025035[n_] := A025035[n] = (3n)!/n!/6^n; z[n_] := z[n] = Sum[Binomial[n, k+k]*(k+k-1)!!*k^(n-k-k), {k, 0, Floor[n/2]}]; r[n_] := r[n] = n!*Sum[Sum[Sum[Sum[StirlingS2[e, d]*a^(d+i)*(a*(a+1)/2)^(n-i-i-e-d-a)/a!/(n-i-i-e-d-a)!/i!/e!/2^i, {a, 0, n-i-i-e-d}], {d, 0, Min[e, n-i-i-e]}], {e, 0, n-i-i}], {i, 0, Floor[n/2]}]; A135429[n_] := A135429[n] = n!*Sum[Sum[A025035[i]*z[j]*r[n-3*i-j]/(3 i)!/j!/(n-3*i-j)!, {j, 0, n-3*i}], {i, 0, Floor[n/3]}]; A135312[n_] := SeriesCoefficient[Exp[x*Exp[x]+x^2/2], {x, 0, n}]*n!; a[n_] := A135429[n]-A135312[n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 05 2014, after Alois P. Heinz*)

Formula

a(n) = A135429(n) - A135312(n).

A218092 Number of transitive reflexive early confluent binary relations R on n labeled elements with max_{x}(|{y : xRy}|) = 2.

Original entry on oeis.org

3, 12, 61, 310, 1821, 11592, 80963, 608832, 4910785, 42159238, 383478987, 3678859158, 37087880753, 391641822540, 4319860660447, 49647399946048, 593217470459313, 7354718987639958, 94445777492433515, 1254196823154143190, 17198114810490326713
Offset: 2

Views

Author

Alois P. Heinz, Oct 20 2012

Keywords

Comments

R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.

References

  • A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.

Links

Crossrefs

Column k=2 of A135313.

Programs

  • Maple
    egf:= exp(x*exp(x)+x^2/2)-exp(x):
a:= n-> n!* coeff(series(egf, x, n+1), x, n):
seq(a(n), n=2..30);
  • Mathematica
    nn=22; Drop[CoefficientList[Series[Exp[x Exp[x]+x^2/2]-Exp[x],{x,0,nn}],x] Range[0,nn]!,2] (* Harvey P. Dale, May 03 2014 *)

Formula

E.g.f.: exp(x*exp(x)+x^2/2)-exp(x).
a(n) = A135312(n) - A000012(n).

A218093 Number of transitive reflexive early confluent binary relations R on n labeled elements with max_{x}(|{y : xRy}|) = 3.

Original entry on oeis.org

13, 106, 1105, 12075, 141533, 1812216, 25188019, 378725365, 6135529675, 106586385708, 1976799958367, 38978490654831, 814024466784025, 17943457752971680, 416183933276776375, 10128962147830237953, 258021086313431979827, 6863916836407264864380
Offset: 3

Views

Author

Alois P. Heinz, Oct 20 2012

Keywords

Comments

R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.

References

  • A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.

Links

Crossrefs

Column k=3 of A135313.

Programs

  • Maple
    egf:= exp(x*exp(x*exp(x)+x^2/2)+x^2/2*exp(x)+x^3/6)-exp(x*exp(x)+x^2/2):
a:= n-> n!* coeff(series(egf, x, n+1), x, n):
seq(a(n), n=3..30);

Formula

E.g.f.: exp(x*exp(x*exp(x)+x^2/2)+x^2/2*exp(x)+x^3/6)-exp(x*exp(x)+x^2/2).
a(n) = A210911(n) - A135312(n).
Showing 1-5 of 5 results.

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