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 1-10 of 19 results. Next

A052880 Expansion of e.g.f.: LambertW(1-exp(x))/(1-exp(x)).

Original entry on oeis.org

1, 1, 4, 26, 243, 2992, 45906, 845287, 18182926, 447797646, 12429760889, 384055045002, 13075708703910, 486430792977001, 19632714343389296, 854503410602781782, 39898063449977239323, 1989371798838577172796, 105503454201101084456182, 5930110732782743218645271
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Comments

A simple grammar.
Also the number of transitive reflexive early confluent binary relations R on n labeled elements. Early confluency means that (xRy and xRz) implies (yRz or zRy) for all x, y, z.

Links

Crossrefs

Row sums of A135313.
Main diagonal of A135302.
Cf. A033917, A053763.

Programs

  • Maple
    spec := [S,{B=Set(Z,1 <= card),S=Set(C),C=Prod(B,S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
# second Maple program:
b:= proc(n, m) option remember; `if`(n=0,
     (m+1)^(m-1), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..27);  # Alois P. Heinz, Jul 15 2022
  • Mathematica
    CoefficientList[Series[-LambertW[-E^x+1]/(E^x-1), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)
f[0, ] = 1; f[k, x_] := f[k, x] = Exp[Sum[x^m/m!*f[k-m, x], {m, 1, k}]];
(* b = A135302 *) b[0, 0] = 1; b[, 0] = 0; b[n, k_] := SeriesCoefficient[ f[k, x], {x, 0, n}]*n!;
a[n_] := b[n, n];
a /@ Range[0, 20] (* Jean-François Alcover, Oct 14 2019 *)
  • PARI
    {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, sum(k=0, m, Stirling2(m, k)*(A+x*O(x^n))^k)*x^m/m!)); n!*polcoeff(A, n)} \\ Paul D. Hanna, Mar 09 2013
  • PARI
    x='x+O('x^30); Vec(serlaplace(-lambertw(-exp(x)+1)/(exp(x)-1))) \\ G. C. Greubel, Feb 19 2018

Formula

a(n) = Sum_{k=0..n} Stirling2(n, k)*(k+1)^(k-1). - Vladeta Jovovic, Nov 12 2003
a(n) ~ sqrt(1+exp(1)) * n^(n-1) / (exp(n-1)*(log(1+exp(1))-1)^(n-1/2)). - Vaclav Kotesovec, Nov 27 2012
E.g.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n/n! * Sum_{k=0..n} Stirling2(n,k) * A(x)^k. - Paul D. Hanna, Mar 09 2013
E.g.f. A(x) satisfies: A(x) = exp((exp(x) - 1)*A(x)). - Ilya Gutkovskiy, Apr 04 2019

Extensions

Edited by Alois P. Heinz, Nov 21 2010

A135313 Triangle of numbers T(n,k) (n>=0, n>=k>=0) of transitive reflexive early confluent binary relations R on n labeled elements where k=max_{x}(|{y : xRy}|), read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 1, 12, 13, 0, 1, 61, 106, 75, 0, 1, 310, 1105, 1035, 541, 0, 1, 1821, 12075, 16025, 11301, 4683, 0, 1, 11592, 141533, 267715, 239379, 137774, 47293, 0, 1, 80963, 1812216, 4798983, 5287506, 3794378, 1863044, 545835, 0, 1, 608832, 25188019, 92374107, 124878033, 105494886, 64432638, 27733869, 7087261
Offset: 0

Views

Author

Alois P. Heinz, Dec 05 2007

Keywords

Comments

R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.
Conjecture: For fixed k>=0, A135313(n+k,n) ~ n! * n^(2*k) / (2^(k+1) * k! * log(2)^(n+k+1)). - Vaclav Kotesovec, Nov 20 2021

Examples

			T(3,3) = 13 because there are 13 relations of the given kind for 3 elements:  (1) 1R2, 2R1, 1R3, 3R1, 2R3, 3R2;  (2) 1R2, 1R3, 2R3, 3R2;  (3) 2R1, 2R3, 1R3, 3R1;  (4) 3R1, 3R2, 1R2, 2R1;  (5) 2R1, 3R1, 2R3, 3R2; (6) 1R2, 3R2, 1R3, 3R1;  (7) 1R3, 2R3, 1R2, 2R1;  (8) 1R2, 2R3, 1R3;  (9) 1R3, 3R2, 1R2;  (10) 2R1, 1R3, 2R3;  (11) 2R3, 3R1, 2R1;  (12) 3R1, 1R2, 3R2;  (13) 3R2, 2R1, 3R1; (the reflexive relationships 1R1, 2R2, 3R3 have been omitted for brevity).
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,    3;
  0,  1,   12,    13;
  0,  1,   61,   106,    75;
  0,  1,  310,  1105,  1035,   541;
  0,  1, 1821, 12075, 16025, 11301, 4683;
  ...

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, A057427, A218092, A218093, A218094, A218095, A218096, A218097, A218098, A218099, A218091.
Main diagonal and lower diagonals give: A000670, A218111, A218112, A218103, A218104, A218105, A218106, A218107, A218108, A218109, A218110.
Row sums are in A052880.
T(2n,n) gives A261238.
Cf. A135302.

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:
tt:= proc(k) option remember;
       unapply((t(k)-t(k-1))(x), x)
     end:
T:= proc(n, k) option remember;
      coeff(series(tt(k)(x), x, n+1), x, n)*n!
    end:
seq(seq(T(n, k), k=0..n), n=0..12);
  • Mathematica
    f[0, ] = 1; f[k, x_] := f[k, x] = Exp[Sum[x^m/m!*f[k - m, x], {m, 1, k}]]; (* a = A135302 *) a[0, 0] = 1; a[, 0] = 0; a[n, k_] := SeriesCoefficient[f[k, x], {x, 0, n}]*n!; t[n_, 0] := a[n, 0]; t[n_, k_] := a[n, k] - a[n, k-1]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 06 2013, after A135302 *)

Formula

T(n,0) = A135302(n,0), T(n,k) = A135302(n,k) - A135302(n,k-1) for k>0.
E.g.f. of column k=0: tt_0(x) = 1, e.g.f. of column k>0: tt_k(x) = t_k(x) -t_{k-1}(x), where t_k(x) = exp (Sum_{m=1..k} x^m/m! * t_{k-m}(x)) if k>=0 and t_k(x) = 0 else.

A135312 Number of transitive reflexive binary relations R on n labeled elements where |{y : xRy}| <= 2 for all x.

Original entry on oeis.org

1, 1, 4, 13, 62, 311, 1822, 11593, 80964, 608833, 4910786, 42159239, 383478988, 3678859159, 37087880754, 391641822541, 4319860660448, 49647399946049, 593217470459314, 7354718987639959, 94445777492433516, 1254196823154143191, 17198114810490326714, 243191242578584519333
Offset: 0

Views

Author

Alois P. Heinz, Dec 05 2007

Keywords

Examples

			a(2) = 4 because there are 4 relations of the given kind for 2 elements: 1R1, 2R2;  1R1, 2R2, 1R2;  1R1, 2R2, 2R1;  1R1, 2R2, 1R2, 2R1.

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 A135302.
Cf. A006882, A000248, A007318.

Programs

  • Maple
    u:= proc(n) option remember; add(binomial(n, i)*(n-i)^i, i=0..n) end:
a:= n-> add(binomial(n, 2*i)*doublefactorial(2*i-1)*u(n-2*i), i=0..iquo(n, 2)):
seq(a(n), n=0..50);
  • Mathematica
    a[n_] := SeriesCoefficient[Exp[x*Exp[x] + x^2/2], {x, 0, n}]*n!; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 04 2014 *)

Formula

a(n) = Sum_{i=0..floor(n/2)} C(n,2*i) * A006882(2*i-1) * A000248(n-2*i).
a(n) = A135302(n,2).
E.g.f.: exp(x*exp(x) + x^2/2).

A210911 Number of transitive reflexive early confluent binary relations R on n labeled elements where |{y : xRy}| <= 3 for all x.

Original entry on oeis.org

1, 1, 4, 26, 168, 1416, 13897, 153126, 1893180, 25796852, 383636151, 6177688914, 106969864696, 1980478817526, 39015578535585, 814416108606566, 17947777613632128, 416233580676722424, 10129555365300697267, 258028441032419619786, 6864011282184757297896
Offset: 0

Views

Author

Alois P. Heinz, Mar 29 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 A135302.

Programs

  • Maple
    gf:= exp(x*exp(x*exp(x)+x^2/2)+x^2/2*exp(x)+x^3/6):
a:= n-> n!* coeff(series(gf,x,n+1), x, n):
seq(a(n), n=0..30);
  • 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, 3], {n, 0, 30} ] (* Jean-François Alcover, Feb 04 2014, after A135302 and Alois P. Heinz *)

Formula

E.g.f.: exp(x*exp(x*exp(x)+x^2/2)+x^2/2*exp(x)+x^3/6).

A210912 Number of transitive reflexive early confluent binary relations R on n labeled elements where |{y : xRy}| <= 4 for all x.

Original entry on oeis.org

1, 1, 4, 26, 243, 2451, 29922, 420841, 6692163, 118170959, 2296688956, 48661358989, 1115587992521, 27499790373121, 725031761113038, 20351018228318061, 605726610363853513, 19050158234570819809, 631097355371645795620, 21961423837720097681425
Offset: 0

Views

Author

Alois P. Heinz, Mar 29 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=4 of A135302.

Programs

  • Maple
    gf:= exp(x *exp(x *exp(x *exp(x)+x^2/2) +x^2/2*exp(x) +x^3/6)
         +x^2/2 *exp(x*exp(x) +x^2/2) +x^3/6 *exp(x) +x^4/24):
a:= n-> n!* coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..30);
  • 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, 4], {n, 0, 30} ] (* Jean-François Alcover, Feb 04 2014, after A135302 and Alois P. Heinz *)

Formula

E.g.f.: exp(x *exp(x *exp(x *exp(x)+x^2/2) +x^2/2*exp(x) +x^3/6) +x^2/2 *exp(x*exp(x) +x^2/2) +x^3/6 *exp(x) +x^4/24).

A210913 Number of transitive reflexive early confluent binary relations R on n labeled elements where |{y : xRy}| <= 5 for all x.

Original entry on oeis.org

1, 1, 4, 26, 243, 2992, 41223, 660220, 11979669, 243048992, 5448497434, 133595966164, 3555887814602, 102064563003898, 3141580135645330, 103198691666336823, 3602725068242695657, 133174089439019869924, 5195463138498447345478, 213295995976349091757666
Offset: 0

Views

Author

Alois P. Heinz, Mar 29 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=5 of A135302.

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:
gf:= t(5)(x):
a:= n-> n!*coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..30);
  • 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, 5], {n, 0, 30} ] (* Jean-François Alcover, Feb 04 2014, after A135302 and Alois P. Heinz *)

Formula

E.g.f.: t_5(x), where t_k(x) = exp (Sum_{m=1..k} x^m/m! * t_{k-m}(x)) for k>=0 and t_k(x) = 0 otherwise.

A210914 Number of transitive reflexive early confluent binary relations R on n labeled elements where |{y : xRy}| <= 6 for all x.

Original entry on oeis.org

1, 1, 4, 26, 243, 2992, 45906, 797994, 15774047, 348543878, 8517326911, 228090873748, 6641805913833, 208882903017855, 7054977212140236, 254641097826922363, 9780088146805724737, 398202474048334638184, 17130262219179411169927, 776303072938412423933278
Offset: 0

Views

Author

Alois P. Heinz, Mar 29 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=6 of A135302.

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:
gf:= t(6)(x):
a:= n-> n!* coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..20);
  • Mathematica
    t[k_] := t[k] = If[k<0, 0, Function[x, Exp[Sum[x^m/m!*t[k-m][x], {m, 1, k}]]]]; gf = t[6][x]; a[n_] := n!*SeriesCoefficient [gf, {x, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 13 2014, translated from Maple *)

Formula

E.g.f.: t_6(x), where t_k(x) = exp (Sum_{m=1..k} x^m/m! * t_{k-m}(x)) for k>=0 and t_k(x) = 0 otherwise.

A210915 Number of transitive reflexive early confluent binary relations R on n labeled elements where |{y : xRy}| <= 7 for all x.

Original entry on oeis.org

1, 1, 4, 26, 243, 2992, 45906, 845287, 17637091, 412976516, 10702355041, 304058582059, 9396887340381, 313853270626962, 11265355519125229, 432420217726582213, 17674492093095982705, 766343475354260380416, 35129831766609666284023, 1697466558811335003294745
Offset: 0

Views

Author

Alois P. Heinz, Mar 29 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=7 of A135302.

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:
gf:= t(7)(x):
a:= n-> n!* coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..30);
  • Mathematica
    t[k_] := t[k] = If[k<0, 0, Function[x, Exp[Sum[x^m/m!*t[k-m][x], {m, 1, k}]]]]; gf = t[7][x]; a[n_] := n!*SeriesCoefficient[gf, {x, 0, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 13 2014, translated from Maple *)

Formula

E.g.f.: t_7(x), where t_k(x) = exp (Sum_{m=1..k} x^m/m! * t_{k-m}(x)) for k>=0 and t_k(x) = 0 otherwise.

A210916 Number of transitive reflexive early confluent binary relations R on n labeled elements where |{y : xRy}| <= 8 for all x.

Original entry on oeis.org

1, 1, 4, 26, 243, 2992, 45906, 845287, 18182926, 440710385, 11876274391, 351546957499, 11330575607067, 394862762014644, 14792903605828298, 592835563146850723, 25306351970600498930, 1146305330627242918543, 54914971513967144548105, 2773947252964889935144249
Offset: 0

Views

Author

Alois P. Heinz, Mar 29 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=8 of A135302.

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:
gf:= t(8)(x):
a:= n-> n!* coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..30);
  • Mathematica
    t[k_] := t[k] = If[k<0, 0, Function[x, Exp[Sum[x^m/m!*t[k-m][x], {m, 1, k}]]]]; gf = t[8][x]; a[n_] := n!*SeriesCoefficient[gf, {x, 0, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 13 2014, translated from Maple *)

Formula

E.g.f.: t_8(x), where t_k(x) = exp (Sum_{m=1..k} x^m/m! * t_{k-m}(x)) for k>=0 and t_k(x) = 0 otherwise.

A210917 Number of transitive reflexive early confluent binary relations R on n labeled elements where |{y : xRy}| <= 9 for all x.

Original entry on oeis.org

1, 1, 4, 26, 243, 2992, 45906, 845287, 18182926, 447797646, 12327513326, 374460094229, 12417692352452, 445937963850159, 17230880407496706, 712587605616915013, 31399448829720502520, 1468521294946336416768, 72650756455913144620677, 3790469182850937732166657
Offset: 0

Views

Author

Alois P. Heinz, Mar 29 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=9 of A135302.

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:
gf:= t(9)(x):
a:= n-> n!*coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..30);
  • Mathematica
    t[k_] := t[k] = If[k<0, 0, Function[x, Exp[Sum[x^m/m!*t[k-m][x], {m, 1, k}]]]]; gf = t[9][x]; a[n_] := n!*SeriesCoefficient[gf, {x, 0, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 13 2014, translated from Maple *)

Formula

E.g.f.: t_9(x), where t_k(x) = exp (Sum_{m=1..k} x^m/m! * t_{k-m}(x)) for k>=0 and t_k(x) = 0 otherwise.
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