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 1-4 of 4 results.

A220833 Erroneous version of A339782.

Original entry on oeis.org

2, 11, 28, 109, 470, 2145, 10300, 51135, 260930, 1359391
Offset: 1

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Author

Keywords

Comments

Number of unrooted non-binary leaf-multi-labeled trees with n leaves on the label set [2].
The reference has a mistake in formula 4.3. Rather than "k*g(n-1,k) + g(n,k) + Sum_{j=1..n-1} g(j,k)*g(n-j,k)" it should be "k*g(n-1,k) + g(n,k) - Sum_{j=1..n-1} g(j,k)*g(n-j,k)". Table 4.3 (A220832, this sequence, A220834, A220835) is consequently also incorrect.

Links

A339779 Array read by antidiagonals: T(n,k) is the number of homeomorphically irreducible leaf colored trees with n leaves of k colors.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 6, 4, 2, 0, 1, 5, 10, 10, 11, 3, 0, 1, 6, 15, 20, 36, 30, 7, 0, 1, 7, 21, 35, 90, 144, 105, 13, 0, 1, 8, 28, 56, 190, 476, 706, 380, 32, 0, 1, 9, 36, 84, 357, 1251, 3034, 3774, 1555, 73, 0, 1, 10, 45, 120, 616, 2814, 9845, 21380, 22140, 6650, 190, 0
Offset: 0

Views

Author

Andrew Howroyd, Dec 16 2020

Keywords

Comments

Homeomorphically irreducible trees are trees without vertices of degree 2. All non-leaf nodes then have degree >= 3.
Not all colors need to be used.
The Johnson reference has a mistake in formula 4.3. In particular, the final term should be subtracted rather than added. Compare with the first formula given here. The table of results given in the reference is consequently also incorrect.

Examples

			Array begins:
============================================================
n\k| 0  1    2      3       4       5        6         7
---+--------------------------------------------------------
0  | 1  1    1      1       1       1        1         1 ...
1  | 0  1    2      3       4       5        6         7 ...
2  | 0  1    3      6      10      15       21        28 ...
3  | 0  1    4     10      20      35       56        84 ...
4  | 0  2   11     36      90     190      357       616 ...
5  | 0  3   30    144     476    1251     2814      5656 ...
6  | 0  7  105    706    3034    9845    26383     61572 ...
7  | 0 13  380   3774   21380   85995   274800    744556 ...
8  | 0 32 1555  22140  163670  812160  3086481   9692480 ...
9  | 0 73 6650 137096 1322960 8092945 36550458 132954360 ...
     ...

Links

Crossrefs

Columns k=1..4 are A007827, A339782, A339783, A339784.
Cf. A319254 (planted), A339649 (degree <= 3), A339780.

Programs

  • PARI
    \\ here R(n,k) is k-th column of A319254.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(v=[k]); for(n=2, n, v=concat(v, EulerT(concat(v, [0]))[n])); v}
U(n, k)={my(g=x*Ser(R(n,k))); Vec(1 + g + k*x*g - g^2)}
{my(T=Mat(vector(9, k, U(8, k-1)~))); for(n=1, #T~, print(T[n, ]))}

Formula

T(n,k) = k*g(n-1,k) + g(n,k) - Sum_{j=1..n-1} g(j,k)*g(n-j,k) for n > 1 where g(n,k) is A319254(n,k).
G.f. of k-th column: 1 + k*x*r(x) + r(x) - r(x)^2 where r(x) is the g.f. of the k-th column of A319254.

A339785 Number of homeomorphically irreducible leaf colored trees with n leaves using exactly 2 colors.

Original entry on oeis.org

0, 1, 2, 7, 24, 91, 354, 1491, 6504, 29711, 139616, 674696, 3328798, 16730955, 85382210, 441571216, 2310003732, 12206975528, 65082858008, 349756996762, 1892980028014, 10310987833049, 56489307860860, 311112321625754, 1721692801914844, 9569930999155801, 53410232801675436
Offset: 1

Views

Author

Andrew Howroyd, Dec 16 2020

Keywords

Links

Crossrefs

Column k=2 of A339780.
Cf. A007827, A339782.

Programs

  • PARI
    my(N=25); (U(N,2)-2*U(N,1))[2..1+N] \\ See A339780 for U(n,k).

Formula

a(n) = A339782(n) - 2*A007827(n).

A339786 Number of homeomorphically irreducible leaf colored trees with n leaves using exactly 3 colors.

Original entry on oeis.org

0, 0, 1, 9, 63, 412, 2673, 17571, 117365, 798819, 5530122, 38908380, 277750749, 2009160864, 14707923021, 108835512411, 813241695330, 6130521151377, 46584949832013, 356571373433217, 2747371943624943, 21296479544449677, 165994877608025730, 1300408539157086640
Offset: 1

Views

Author

Andrew Howroyd, Dec 18 2020

Keywords

Links

Crossrefs

Column k=3 of A339780.
Cf. A007827, A339782, A339783, A339785.

Programs

  • PARI
    my(N=25); (U(N,3) - 3*U(N,2) + 3*U(N,1))[2..1+N] \\ See A339780 for U(n, k).

Formula

a(n) = A339783(n) - 3*A339782(n) + 3*A007827(n).
Showing 1-4 of 4 results.

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