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A319539
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Array read by antidiagonals: T(n,k) is the number of binary rooted trees with n leaves of k colors and all non-leaf nodes having out-degree 2.
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12
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1, 2, 1, 3, 3, 1, 4, 6, 6, 2, 5, 10, 18, 18, 3, 6, 15, 40, 75, 54, 6, 7, 21, 75, 215, 333, 183, 11, 8, 28, 126, 495, 1260, 1620, 636, 23, 9, 36, 196, 987, 3600, 8010, 8208, 2316, 46, 10, 45, 288, 1778, 8568, 28275, 53240, 43188, 8610, 98, 11, 55, 405, 2970, 17934, 80136, 232500, 366680, 232947, 32763, 207
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OFFSET
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1,2
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COMMENTS
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Not all k colors need to be used. The total number of nodes will be 2n-1.
See table 2.1 in the Johnson reference.
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LINKS
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FORMULA
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T(1,k) = k.
T(n,k) = (1/2)*([n mod 2 == 0]*T(n/2,k) + Sum_{j=1..n-1} T(j,k)*T(n-j,k)).
G.f. of k-th column R(x) satisfies R(k) = k*x + (R(x)^2 + R(x^2))/2.
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EXAMPLE
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Array begins:
===========================================================
n\k| 1 2 3 4 5 6 7
---+-------------------------------------------------------
1 | 1 2 3 4 5 6 7 ...
2 | 1 3 6 10 15 21 28 ...
3 | 1 6 18 40 75 126 196 ...
4 | 2 18 75 215 495 987 1778 ...
5 | 3 54 333 1260 3600 8568 17934 ...
6 | 6 183 1620 8010 28275 80136 194628 ...
7 | 11 636 8208 53240 232500 785106 2213036 ...
8 | 23 2316 43188 366680 1979385 7960638 26037431 ...
9 | 46 8610 232947 2590420 17287050 82804806 314260765 ...
...
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MAPLE
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A:= proc(n, k) option remember; `if`(n<2, k*n, `if`(n::odd, 0,
(t-> t*(1-t)/2)(A(n/2, k)))+add(A(i, k)*A(n-i, k), i=1..n/2))
end:
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MATHEMATICA
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A[n_, k_] := A[n, k] = If[n<2, k*n, If[OddQ[n], 0, (#*(1-#)/2&)[A[n/2, k]]] + Sum[A[i, k]*A[n-i, k], {i, 1, n/2}]];
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PROG
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(PARI) \\ here R(n, k) gives k-th column as a vector.
R(n, k)={my(v=vector(n)); v[1]=k; for(n=2, n, v[n]=sum(j=1, (n-1)\2, v[j]*v[n-j]) + if(n%2, 0, binomial(v[n/2]+1, 2))); v}
{my(T=Mat(vector(8, k, R(8, k)~))); for(n=1, #T~, print(T[n, ]))}
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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