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User: Stephan Beyer

Stephan Beyer has authored 2 sequences.

A271362 Number T(n,k) of series-reduced free trees with n nodes of which exactly k>=3 are leaves, k+1 <= n <= 2k-2.

1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 4, 3, 1, 4, 6, 3, 1, 2, 10, 9, 4, 1, 8, 17, 12, 4, 1, 4, 22, 30, 16, 5, 1, 15, 47, 44, 20, 5, 1, 6, 53, 91, 67, 25, 6, 1, 32, 127, 158, 91, 30, 6, 1, 11, 121, 282, 258, 126

Offset: 4

Stephan Beyer, Apr 05 2016

The length of row n is floor((n-2)/2).

			    Irregular triangle begins
    n \ k 3   4   5   6   7  8
     4     1;
     5     1;
     6     1,  1;
     7     1,  1;
     8     1,  2,  1;
     9     2,  2,  1;
    10     2,  4,  3,  1;
    11     4,  6,  3,  1;
    12     2, 10,  9,  4, 1;
    13     8, 17, 12,  4, 1;
    14     4, 22, 30, 16, 5, 1;
    15    15, 47, 44, 20, 5, 1;
    ...

Washington Bomfim, Table of n, a(n) for n = 4..199
B. D. McKay, Lists of Trees sorted by diameter and Homeomorphically irreducible trees, with <= 22 nodes.

Transpose of A271205.

Cf. A000014 (row sums), A345971.

\\ using files hitree4.txt etc from McKay.
nL(n, Tr) = { my(E = strsplit(Tr, "  "), u_v, Deg = vectorsmall(n));
for(j = 1, n-1, u_v = strsplit(E[j], " "); u_v = eval(u_v);
   Deg[ u_v[1]+1 ]++; Deg[ u_v[2]+1 ]++); sum(v = 1, n, Deg[v] == 1)
};
Rows(r1, r2) = {my(F, C, nF); for(n = r1, r2,
F = readstr(Str("hitree", n, ".txt")); C = vectorsmall(n-1);
for(i = 1, #F, nF = nL(n, F[i]); C[nF]++ );
print1(n" "); for(i=1, #C, if(C[i] > 0, print1(C[i]", "))); print() )
}; \\ Washington Bomfim, Jul 09 2021

T(n,k) = A271205(k,n).

A271205 Number T(m,n) of series-reduced free trees with n nodes of which exactly m >= 3 are leaves, m+1 <= n <= 2m-2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 4, 4, 2, 1, 3, 6, 10, 8, 4, 1, 3, 9, 17, 22, 15, 6, 1, 4, 12, 30, 47, 53, 32, 11, 1, 4, 16, 44, 91, 127, 121, 66, 18, 1, 5, 20, 67, 158, 282, 346, 292, 142, 37, 1, 5, 25, 91, 258, 539, 841, 921, 688, 306, 66, 1, 6, 30, 126, 397, 978, 1804, 2498, 2456, 1662, 672, 135, 1, 6, 36, 163, 588, 1636, 3550, 5856, 7260, 6489, 3978, 1483, 265, 1, 7, 42, 213, 838, 2638, 6495, 12554, 18636, 20946, 17082, 9629, 3316, 552, 1, 8

Offset: 3

Stephan Beyer, Apr 01 2016

The sequence of row sums a(m) = Sum_{n} T(m,n) is A007827.

The sequence of column sums a(n) = Sum_{m} T(m,n) is A000014.

			m\n | 3 4 5 6 7 8 9 10 11 12 13 14 15 16  17  18 19 20
-------------------------------------------------------
3   | . 1 . . . . .  .  .  .  .  .  .  .   .   .  .  .
4   | . . 1 1 . . .  .  .  .  .  .  .  .   .   .  .  .
5   | . . . 1 1 1 .  .  .  .  .  .  .  .   .   .  .  .
6   | . . . . 1 2 2  2  .  .  .  .  .  .   .   .  .  .
7   | . . . . . 1 2  4  4  2  .  .  .  .   .   .  .  .
8   | . . . . . . 1  3  6 10  8  4  .  .   .   .  .  .
9   | . . . . . . .  1  3  9 17 22 15  6   .   .  .  .
10  | . . . . . . .  .  1  4 12 30 47 53  32  11  .  .
11  | . . . . . . .  .  .  1  4 16 44 91 127 121 66 18

Stephan Beyer, Python code to generate the sequence on GitHub Gist

Cf. A000014, A007827.

Transpose of A271362.

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User: Stephan Beyer

A271362 Number T(n,k) of series-reduced free trees with n nodes of which exactly k>=3 are leaves, k+1 <= n <= 2k-2.

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