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A319376 Triangle read by rows: T(n,k) is the number of series-reduced rooted trees with n leaves of exactly k colors. 6
1, 1, 1, 2, 6, 4, 5, 30, 51, 26, 12, 146, 474, 576, 236, 33, 719, 3950, 8572, 8060, 2752, 90, 3590, 31464, 108416, 175380, 134136, 39208, 261, 18283, 245916, 1262732, 3124650, 4014348, 2584568, 660032, 766, 94648, 1908858, 14047288, 49885320, 95715728, 101799712, 56555904, 12818912 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

LINKS

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

FORMULA

T(n,k) = Sum_{i=1..k} (-1)^(k-i)*binomial(k,i)*A319254(n,i).

Sum_{k=1..n} k * T(n,k) = A326396(n). - Alois P. Heinz, Sep 11 2019

EXAMPLE

Triangle begins:

    1;

    1,     1;

    2,     6,      4;

    5,    30,     51,      26;

   12,   146,    474,     576,     236;

   33,   719,   3950,    8572,    8060,   2752;

   90,  3590,  31464,  108416,  175380,  134136,   39208;

  261, 18283, 245916, 1262732, 3124650, 4014348, 2584568, 660032;

  ...

MAPLE

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,

      add(binomial(A(i, k)+j-1, j)*b(n-i*j, i-1, k), j=0..n/i)))

    end:

A:= (n, k)-> `if`(n<2, n*k, b(n, n-1, k)):

T:= (n, k)-> add(A(n, k-j)*(-1)^j*binomial(k, j), j=0..k-1):

seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Sep 18 2018

MATHEMATICA

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[A[i, k] + j - 1, j] b[n - i j, i - 1, k], {j, 0, n/i}]]];

A[n_, k_] := If[n < 2, n k, b[n, n - 1, k]];

T[n_, k_] := Sum[(-1)^(k - i)*Binomial[k, i]*A[n, i], {i, 1, k}];

Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-Fran├žois Alcover, Sep 24 2019, after Alois P. Heinz *)

PROG

(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}

M(n)={my(v=vector(n, k, R(n, k)~)); Mat(vector(n, k, sum(i=1, k, (-1)^(k-i)*binomial(k, i)*v[i])))}

{my(T=M(10)); for(n=1, #T~, print(T[n, ][1..n]))}

CROSSREFS

Columns k=1..2 are A000669, A319377.

Main diagonal is A000311.

Row sums are A316651.

Cf. A141610, A242249, A255517, A256064, A256068, A319254, A319541, A326396.

Sequence in context: A182505 A010465 A065630 * A110633 A240232 A119250

Adjacent sequences:  A319373 A319374 A319375 * A319377 A319378 A319379

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Sep 17 2018

STATUS

approved

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Last modified March 30 16:16 EDT 2020. Contains 333127 sequences. (Running on oeis4.)