A144215 Triangle read by rows: T(n,k) is the number of forests on n unlabeled nodes with all nodes of degree <= k (n>=1, 0 <= k <= n-1).
1, 1, 2, 1, 2, 3, 1, 3, 5, 6, 1, 3, 7, 9, 10, 1, 4, 11, 17, 19, 20, 1, 4, 15, 28, 34, 36, 37, 1, 5, 22, 52, 67, 73, 75, 76, 1, 5, 30, 90, 129, 144, 150, 152, 153, 1, 6, 42, 170, 264, 305, 320, 326, 328, 329, 1, 6, 56, 310, 542, 645, 686, 701, 707, 709, 710
Offset: 1
Examples
Triangle begins: 1 1 2 1 2 3 1 3 5 6 1 3 7 9 10 1 4 11 17 19 20 1 4 15 28 34 36 37 ... From _Andrew Howroyd_, Dec 18 2020: (Start) Formatted as an array to show the full columns: ======================================================== n\k | 0 1 2 3 4 5 6 7 8 9 10 -----+-------------------------------------------------- 1 | 1 1 1 1 1 1 1 1 1 1 1 ... 2 | 1 2 2 2 2 2 2 2 2 2 2 ... 3 | 1 2 3 3 3 3 3 3 3 3 3 ... 4 | 1 3 5 6 6 6 6 6 6 6 6 ... 5 | 1 3 7 9 10 10 10 10 10 10 10 ... 6 | 1 4 11 17 19 20 20 20 20 20 20 ... 7 | 1 4 15 28 34 36 37 37 37 37 37 ... 8 | 1 5 22 52 67 73 75 76 76 76 76 ... 9 | 1 5 30 90 129 144 150 152 153 153 153 ... 10 | 1 6 42 170 264 305 320 326 328 329 329 ... 11 | 1 6 56 310 542 645 686 701 707 709 710 ... 12 | 1 7 77 600 1161 1431 1536 1577 1592 1598 1600 ... (End)
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
- Rebecca Neville, Graphs whose vertices are forests with bounded degree, Graph Theory Notes of New York, LIV (2008), 12-21. [Wayback Machine link]
Programs
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PARI
\\ Here V(n,k) gives column k of A144528. EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)} MSet(p,k)={my(n=serprec(p,x)-1); if(min(k,n)<1, 1 + O(x*x^n), polcoef(exp( sum(i=1, min(k,n), (y^i + O(y*y^k))*subst(p + O(x*x^(n\i)), x, x^i)/i ))/(1-y + O(y*y^k)), k, y))} V(n,k)={my(g=1+O(x)); for(n=2, n, g=x*MSet(g, k-1)); Vec(1 + x*MSet(g, k) + (subst(g, x, x^2) - g^2)/2)} M(n, m=n)={Mat(vector(m, k, EulerT(V(n,k-1)[2..1+n])~))} { my(T=M(12)); for(n=1, #T~, print(T[n, 1..n])) } \\ Andrew Howroyd, Dec 18 2020
Formula
Column k is Euler transform of column k of A144528. - Andrew Howroyd, Dec 18 2020
Extensions
Terms a(29) and beyond from Andrew Howroyd, Dec 18 2020