login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A339067 Triangle read by rows: T(n,k) is the number of linear forests with n nodes and k rooted trees. 11
1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 9, 12, 9, 4, 1, 20, 30, 25, 14, 5, 1, 48, 74, 69, 44, 20, 6, 1, 115, 188, 186, 133, 70, 27, 7, 1, 286, 478, 503, 388, 230, 104, 35, 8, 1, 719, 1235, 1353, 1116, 721, 369, 147, 44, 9, 1, 1842, 3214, 3651, 3168, 2200, 1236, 560, 200, 54, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

T(n,k) is the number of trees with n nodes rooted at two noninterchangeable nodes at a distance k-1 from each other.

LINKS

Alois P. Heinz, Rows n = 1..200, flattened

FORMULA

G.f. of k-th column: t(x)^k where t(x) is the g.f. of A000081.

Sum_{k=1..n} k * T(n,k) = A038002(n). - Alois P. Heinz, Dec 04 2020

EXAMPLE

Triangle begins:

    1;

    1,    1;

    2,    2,    1;

    4,    5,    3,    1;

    9,   12,    9,    4,   1;

   20,   30,   25,   14,   5,   1;

   48,   74,   69,   44,  20,   6,   1;

  115,  188,  186,  133,  70,  27,   7,  1;

  286,  478,  503,  388, 230, 104,  35,  8, 1;

  719, 1235, 1353, 1116, 721, 369, 147, 44, 9, 1;

  ...

MAPLE

b:= proc(n) option remember; `if`(n<2, n, (add(add(d*b(d),

      d=numtheory[divisors](j))*b(n-j), j=1..n-1))/(n-1))

    end:

T:= proc(n, k) option remember; `if`(k=1, b(n), (t->

      add(T(j, t)*T(n-j, k-t), j=1..n-1))(iquo(k, 2)))

    end:

seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Dec 04 2020

MATHEMATICA

b[n_] := b[n] = If[n < 2, n, (Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n - j], {j, 1, n - 1}])/(n - 1)];

T[n_, k_] := T[n, k] = If[k == 1, b[n], With[{t = Quotient[k, 2]}, Sum[T[j, t]*T[n - j, k - t], {j, 1, n - 1}]]];

Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-Fran├žois Alcover, Jan 03 2021, after Alois P. Heinz *)

PROG

(PARI) \\ TreeGf is A000081.

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(t=TreeGf(max(0, n+1-k))); Vec(t^k, -n)}

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..6 are A000081, A000106, A000242, A000300, A000343, A000395.

Row sums are A000107.

T(2n-1,n) gives A339440.

Cf. A033185, A038002, A217781, A339428.

Sequence in context: A105306 A183191 A273713 * A322329 A064189 A273897

Adjacent sequences:  A339064 A339065 A339066 * A339068 A339069 A339070

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Dec 03 2020

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 24 05:05 EDT 2021. Contains 346273 sequences. (Running on oeis4.)