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 1-3 of 3 results.

A338706 Number of 2-linear trees on n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 3, 10, 24, 56, 114, 224, 411, 733, 1252, 2091, 3393, 5408, 8440, 12982, 19650, 29388, 43394, 63430, 91754, 131584, 187057, 263932, 369624, 514253, 710838, 976876, 1334828, 1814492, 2454011, 3303436, 4426627, 5906599, 7848883, 10389557
Offset: 1

Views

Author

N. J. A. Sloane, Nov 05 2020, using data supplied by Eric Wityk

Keywords

Comments

A k-linear tree is a tree with exactly k vertices of degree 3 or higher all of which lie on a path. - Andrew Howroyd, Dec 17 2020
Empirically the partial sums of A000147. - Sean A. Irvine, Jul 11 2022

Examples

			The a(6) = 1 tree is:
         o   o
         |   |
     o---o---o---o

Links

Crossrefs

Column k=2 of A380363 and A238415.
Cf. A000041, A000147, A130131, A338707, A338708.

Programs

  • PARI
    seq(n)=my(p=1/(eta(x + O(x^(n-3))))); Vec(((x*(p - 1/(1-x)))^2 + x^2*(subst(p,x,x^2) - 1/(1-x^2)))/(2*(1-x)), -n) \\ Andrew Howroyd, Dec 17 2020

Formula

G.f.: ((x*(P(x) - 1/(1-x)))^2 + x^2*(P(x^2) - 1/(1-x^2)))/(2*(1-x)) where P(x) is the g.f. of A000041. - Andrew Howroyd, Dec 17 2020

Extensions

Terms a(31) and beyond from Andrew Howroyd, Dec 17 2020

A338707 Number of 3-linear trees on n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 5, 22, 74, 219, 576, 1394, 3150, 6733, 13744, 26969, 51185, 94323, 169453, 297533, 512006, 865050, 1437739, 2353756, 3801041, 6060918, 9552826, 14894428, 22991659, 35159606, 53299703, 80137271, 119563216, 177091225, 260504790, 380720841
Offset: 1

Views

Author

N. J. A. Sloane, Nov 05 2020, using data supplied by Eric Wityk

Keywords

Comments

A k-linear tree is a tree with exactly k vertices of degree 3 or higher all of which lie on a path. - Andrew Howroyd, Dec 17 2020

Links

Crossrefs

Column k=3 of A380363 and A238415.
Cf. A000041, A130131, A338706, A338708.

Programs

  • PARI
    seq(n)={my(p=1/(eta(x + O(x^(n-5))))); Vec(x^3*(p-1)*((p - 1/(1-x))^2/(1-x)^2 + (subst(p,x,x^2) - 1/(1-x^2))/(1-x^2))/2, -n)} \\ Andrew Howroyd, Dec 17 2020

Formula

G.f.: x^3*(P(x)-1)*((P(x) - 1/(1-x))^2/(1-x)^2 + (P(x^2) - 1/(1-x^2))/(1-x^2))/2 where P(x) is the g.f. of A000041. - Andrew Howroyd, Dec 17 2020

Extensions

Terms a(31) and beyond from Andrew Howroyd, Dec 17 2020

A380363 Triangle read by rows: T(n,k) is the number of linear trees with n vertices and k vertices of degree >= 3, 0 <= k <= max(0, floor(n/2)-1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 7, 3, 1, 11, 10, 1, 1, 17, 24, 5, 1, 25, 56, 22, 1, 1, 36, 114, 74, 6, 1, 50, 224, 219, 37, 1, 1, 70, 411, 576, 158, 8, 1, 94, 733, 1394, 591, 58, 1, 1, 127, 1252, 3150, 1896, 304, 9, 1, 168, 2091, 6733, 5537, 1342, 82, 1
Offset: 0

Views

Author

Andrew Howroyd, Jan 26 2025

Keywords

Comments

A linear tree is a tree with all vertices of degree > 2 belonging to a single path. These are equinumerous with lobster graphs. All trees having at most 3 vertices of degree > 2 are linear trees.

Examples

			Triangle begins:
  1;
  1;
  1;
  1;
  1,   1;
  1,   2;
  1,   4,    1;
  1,   7,    3;
  1,  11,   10,    1;
  1,  17,   24,    5;
  1,  25,   56,   22,    1;
  1,  36,  114,   74,    6;
  1,  50,  224,  219,   37,   1;
  1,  70,  411,  576,  158,   8;
  1,  94,  733, 1394,  591,  58, 1;
  1, 127, 1252, 3150, 1896, 304, 9;
  ...

Links

Crossrefs

Columns 0..4 are A000012, A004250(n-1), A338706, A338707, A338708.
Row sums are A130131.
Cf. A238415 (initial columns same up to k=3).

Programs

  • PARI
    G(n,y)={my(p=1/eta(x + O(x^n)), p2=1/eta(x^2 + O(x^n)),
  g1=(p - 1/(1-x))^2/((1 - x)*(1 - x*y*(p-1)/(1-x))),
  g2=(p2 - 1/(1-x^2))*(1 + x + x*y*(p-1))/((1 - x^2)*(1 - x^2*y^2*(p2-1)/(1-x^2))) );
  x^2*y^2*(g1 + g2)/2 + x*y*(p - 1/((1 + x)*(1 - x)^2)) + 1/(1-x)
}
T(n)=[Vecrev(p) | p<-Vec(G(n,y))]
{my(A=T(15)); for(i=1, #A, print(A[i]))}
Showing 1-3 of 3 results.

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

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