A330763 -id:A330763 - cp's OEIS frontend

A330762 Triangle read by rows: T(n,k) is the number of series-reduced rooted trees whose leaves are multisets of colors with a total of n elements using exactly k colors.

1, 2, 2, 4, 12, 8, 11, 67, 114, 58, 30, 376, 1230, 1496, 612, 96, 2174, 12038, 26156, 24570, 8374, 308, 12792, 113028, 389968, 630300, 481284, 140408, 1052, 76972, 1043355, 5363331, 13259870, 17008218, 10930150, 2785906, 3648, 471260, 9574934, 70524256, 250201560, 479284952, 508811114, 282141552, 63830764

Offset: 1

Andrew Howroyd, Dec 29 2019

			Triangle begins:
     1;
     2,     2;
     4,    12,       8;
    11,    67,     114,      58;
    30,   376,    1230,    1496,      612;
    96,  2174,   12038,   26156,    24570,     8374;
   308, 12792,  113028,  389968,   630300,   481284,   140408;
  1052, 76972, 1043355, 5363331, 13259870, 17008218, 10930150, 2785906;
  ...
The T(3,2) = 12 trees are: (122), (112), ((1)(22)), ((1)(12)), ((2)(12)), ((2)(11)), ((1)(2)(2)), ((1)(1)(2)), ((1)((2)(2))), ((1)((1)(2))), ((2)((1)(2))), ((2)((1)(1))).

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

Column 1 is A141268.
Main diagonal is A005804.
Row sums are A330469.
Cf. A330763 (leaves are sets).

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(v=[]); for(n=1, n, v=concat(v, EulerT(concat(v, [binomial(n+k-1, k-1)]))[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])) }

A330764 Number of series-reduced rooted trees whose leaves are sets with a total of n elements covering an initial interval of positive integers.

Original entry on oeis.org

1, 3, 18, 194, 2944, 57959, 1398858, 39981994, 1320143478, 49439258516, 2070409961552, 95867076538834, 4863079990663528, 268198764863998103, 15977057268090388836, 1022415045656417706598, 69946606996018140613292, 5094427098628436561252367, 393558075509405403487404506

Offset: 1

Andrew Howroyd, Dec 29 2019

nonn

			The a(3) = 18 trees:
  (123)          ((1)(12))       ((1)(1)(1))
  ((1)(23))      ((2)(12))       ((1)((1)(1)))
  ((2)(13))      ((1)(2)(2))
  ((3)(12))      ((1)(1)(2))
  ((1)(2)(3))    ((1)((2)(2)))
  ((1)((2)(3)))  ((1)((1)(2)))
  ((2)((1)(3)))  ((2)((1)(2)))
  ((3)((1)(2)))  ((2)((1)(1)))

Andrew Howroyd, Table of n, a(n) for n = 1..200

Row sums of A330763.

Cf. A330469 (leaves are multisets).

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(v=[]); for(n=1, n, v=concat(v, EulerT(concat(v, [binomial(k, n)]))[n])); v}
seq(n)={sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))}

cp's OEIS Frontend

A330762 Triangle read by rows: T(n,k) is the number of series-reduced rooted trees whose leaves are multisets of colors with a total of n elements using exactly k colors.

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