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A332649
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Array read by antidiagonals: T(n,k) is the number of unlabeled k-gonal cacti having n polygons.
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6
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 4, 6, 1, 1, 1, 1, 3, 7, 8, 11, 1, 1, 1, 1, 4, 8, 25, 19, 23, 1, 1, 1, 1, 4, 13, 31, 88, 48, 47, 1, 1, 1, 1, 5, 14, 67, 132, 366, 126, 106, 1, 1, 1, 1, 5, 20, 80, 372, 636, 1583, 355, 235, 1
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OFFSET
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0,14
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COMMENTS
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The number of nodes will be n*(k-1) + 1.
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LINKS
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EXAMPLE
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Array begins:
======================================================
n\k | 1 2 3 4 5 6 7 8 9
----+-------------------------------------------------
0 | 1 1 1 1 1 1 1 1 1 ...
1 | 1 1 1 1 1 1 1 1 1 ...
2 | 1 1 1 1 1 1 1 1 1 ...
3 | 1 2 2 3 3 4 4 5 5 ...
4 | 1 3 4 7 8 13 14 20 22 ...
5 | 1 6 8 25 31 67 80 143 165 ...
6 | 1 11 19 88 132 372 504 1093 1391 ...
7 | 1 23 48 366 636 2419 3659 9722 13485 ...
8 | 1 47 126 1583 3280 16551 28254 91391 138728 ...
...
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PROG
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(PARI) \\ here R(n, k) is column k+1 of A332648.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(v=[]); for(n=1, n, my(g=1+x*Ser(v)); v=EulerT(Vec((g^k + g^(k%2)*subst(g^(k\2), x, x^2))/2))); concat([1], v)}
U(n, k)={my(p=Ser(R(n, k-1))); my(g(d)=subst(p + O(x*x^(n\d)), x, x^d)); Vec(g(1) + x*sumdiv(k, d, eulerphi(d)*g(d)^(k/d))/(2*k) - x*(g(1)^k)/2 + x*if(k%2==0, g(2)^(k/2) - g(1)^2*g(2)^(k/2-1))/4)}
T(n)={Mat(concat([vectorv(n+1, i, 1)], vector(n, k, Col(U(n, k+1)))))}
{ my(A=T(8)); for(n=1, #A, print(A[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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