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proposed

Andrew Howroyd, <a href="/A330762/b330762.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)

Andrew Howroyd, <a href="/A330762/b330762.txt">Table of n, a(n) for n = 1..1275</a>

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editing

proposed

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proposed

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))).

Row sums are A330469Column 1 is A141268.

Row sums are A330469.

Cf. A330763 (leaves are sets).

Main diagonal is A005804.

allocated for Andrew HowroydTriangle 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

1,2

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;

...

(PARI)

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])) }

Row sums are A330469.

allocated

nonn,tabl

Andrew Howroyd, Dec 29 2019

approved

editing