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.

User: Karen A. Yeats

Karen A. Yeats's wiki page.

Karen A. Yeats has authored 3 sequences.

A332735 Numbers of graphs which are double triangle descendants of K_5 with four more vertices than triangles.

Original entry on oeis.org

1, 6, 15, 34, 61, 106, 162, 246, 342, 477, 626, 825, 1039, 1314, 1606, 1970, 2352, 2817, 3302, 3881, 4481, 5186, 5914, 6758, 7626, 8621, 9642, 10801, 11987, 13322, 14686, 16210, 17764, 19489, 21246, 23185, 25157, 27322, 29522, 31926, 34366, 37021, 39714, 42633, 45591, 48786
Offset: 9

Views

Author

Karen A. Yeats, Feb 21 2020

Keywords

Comments

See Laradji, Mishna, Yeats paper for definition of double triangle descendants.

Links

Crossrefs

Double triangle descendants of K_5 with three more vertices than triangles is A007980. Double triangle descendants of K_5 with two more vertices than triangles is A008619. Double triangle descendants of K_5 with one more vertex than triangles is A000007. Double triangle descendants of K_5 with the same number of vertices as triangles is A000012.

Formula

G.f.: x^9*(1 + 4*x + 3*x^2 + 6*x^3 + 3*x^4 + 4*x^5 + 4*x^7 - 3*x^8 + 3*x^9 - x^10 + x^11)/((1 - x)^4*(1 + x)^2*(1 + x^2)). See Laradji, Mishna, Yeats paper for proof.

A116379 Number of ternary rooted identity (distinct subtrees) trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 12, 25, 52, 113, 245, 542, 1205, 2707, 6113, 13907, 31780, 73010, 168399, 389991, 906231, 2112742, 4939689, 11580640, 27216387, 64110091, 151334814, 357938832, 848153045, 2013190671, 4786210412, 11396004660, 27172368314, 64875527649
Offset: 1

Views

Author

Karen A. Yeats, Feb 06 2006

Keywords

Comments

It is not known if these trees have the asymptotic form C rho^{-n} n^{-3/2}, whereas the identity binary trees, A063895, do, see the Jason P. Bell et al. reference.

Links

Crossrefs

Cf. A004111, A063895, A116380.

Programs

  • C
    #include  using namespace GiNaC; int main(){ int i, order = 40; symbol x("x"); ex T = x; for (i=0; i
  • Maple
    A:= proc(n) option remember; local T; if n<=1 then x else T:= unapply(A(n-1), x); convert(series(x* (1+T(x)+ T(x)^2/2- T(x^2)/2+ T(x)^3/6- T(x)*T(x^2)/2+ T(x^3)/3), x, n+1), polynom) fi end: a:= n-> coeff(A(n), x, n): seq(a(n), n=1..40);  # Alois P. Heinz, Aug 22 2008
  • Mathematica
    A[n_] := A[n] = If[n <= 1, x, T[y_] = A[n-1] /. x -> y; Normal[Series[y*(1+T[y]+T[y]^2/2-T[y^2]/2+T[y]^3/6-T[y]*T[y^2]/2+T[y^3]/3), {y, 0, n+1}]] /. y -> x] ; a[n_] := Coefficient[A[n], x, n]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Feb 13 2014, after Maple *)

Formula

G.f. satisfies: A(x) = x(1+A(x)+A(x)^2/2-A(x^2)/2+A(x)^3/6-A(x)A(x^2)/2+A(x^3)/3), that is A(x) = x(1+Set_{<=3}(A)(x)).

A116380 Number of quaternary rooted identity (distinct subtrees) trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 12, 25, 52, 113, 247, 548, 1226, 2770, 6298, 14419, 33183, 76760, 178327, 415960, 973693, 2286781, 5386573, 12723097, 30127465, 71506140, 170081575, 405359177, 967899981, 2315131955, 5546597838, 13308818691, 31979667219, 76947325788
Offset: 1

Views

Author

Karen A. Yeats, Feb 06 2006

Keywords

Comments

It is not known if these trees have the asymptotic form C rho^{-n} n^{-3/2}, whereas the identity binary trees, A063895, do, see the Jason P. Bell et al. reference.

Links

Crossrefs

Cf. A004111, A063895, A116379.

Programs

  • C
    #include  using namespace GiNaC; int main(){ int i, order=40; symbol x("x"); ex T; for (i=0; i
  • Maple
    A:= proc(n) option remember; local T; if n<=1 then x else T:= unapply(A(n-1), x); convert(series(x* (1+T(x)+ T(x)^2/2- T(x^2)/2+ T(x)^3/6- T(x)*T(x^2)/2+ T(x^3)/3+ T(x)^4/24- T(x)^2* T(x^2)/4+ T(x)* T(x^3)/3+ T(x^2)^2/8- T(x^4)/4), x,n+1), polynom) fi end: a:= n-> coeff(A(n),x,n): seq(a(n), n=1..40); # Alois P. Heinz, Aug 22 2008
  • Mathematica
    A[n_] := A[n] = If[n <= 1, x, T[y_] = A[n-1] /. x -> y; Normal[Series[y*(1+T[y]+T[y]^2/2-T[y^2]/2+T[y]^3/6-T[y]*T[y^2]/2+T[y^3]/3+T[y]^4/24-T[y]^2*T[y^2]/4+T[y]*T[y^3]/3+T[y^2]^2/8-T[y^4]/4), {y, 0, n+1}]] /. y -> x]; a[n_] := Coefficient[A[n], x, n]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Feb 13 2014, after Maple *)

Formula

G.f. satisfies: A(x) = x(1 + A(x) + A(x)^2/2-A(x^2)/2 + A(x)^3/6-A(x)A(x^2)/2+A(x^3)/3 + A(x)^4/24-A(x)^2A(x^2)/4+A(x)A(x^3)/3+A(x^2)^2/8-A(x^4)/4), that is A(x) = x(1+Set_{<=4}(A)(x)).

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.