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A339303 Triangle read by rows: T(n,k) is the number of unoriented linear forests with n nodes and k rooted trees. 4
1, 1, 1, 2, 1, 1, 4, 3, 2, 1, 9, 6, 6, 2, 1, 20, 16, 15, 8, 3, 1, 48, 37, 41, 22, 12, 3, 1, 115, 96, 106, 69, 38, 15, 4, 1, 286, 239, 284, 194, 124, 52, 20, 4, 1, 719, 622, 750, 564, 377, 189, 77, 24, 5, 1, 1842, 1607, 2010, 1584, 1144, 618, 292, 100, 30, 5, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Linear forests (A339067) are considered up to reversal of the linear order.

T(n,k) is the number of unlabeled trees on n nodes rooted at two indistinguishable nodes at distance k-1 from each other.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

FORMULA

G.f of column k: (r(x)^k + r(x)^(k mod 2)*r(x^2)^floor(k/2))/2 where r(x) is the g.f. of A000081.

EXAMPLE

Triangle read by rows:

    1;

    1,   1;

    2,   1,   1;

    4,   3,   2,   1;

    9,   6,   6,   2,   1;

   20,  16,  15,   8,   3,   1;

   48,  37,  41,  22,  12,   3,  1;

  115,  96, 106,  69,  38,  15,  4,  1;

  286, 239, 284, 194, 124,  52, 20,  4, 1;

  719, 622, 750, 564, 377, 189, 77, 24, 5, 1;

  ...

PROG

(PARI) \\ TreeGf is A000081 as g.f.

TreeGf(N) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}

ColSeq(n, k)={my(r=TreeGf(max(0, n+1-k))); Vec(r^k + r^(k%2)*subst(r, x, x^2)^(k\2), -n)/2}

M(n, m=n)=Mat(vector(m, k, ColSeq(n, k)~))

{ my(T=M(12)); for(n=1, #T~, print(T[n, 1..n])) }

CROSSREFS

Columns 1..4 are A000081, A027852, A280788(n-3), A339302.

Row sums are A303840(n+2).

Row sums excluding the first column are A303833.

Cf. A339067.

Sequence in context: A332056 A074744 A341474 * A010360 A262357 A112744

Adjacent sequences:  A339300 A339301 A339302 * A339304 A339305 A339306

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Dec 04 2020

STATUS

approved

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Last modified March 4 23:52 EST 2021. Contains 341812 sequences. (Running on oeis4.)