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%I #20 Feb 04 2023 20:57:57
%S 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,
%T 5,22,52,67,73,75,76,1,5,30,90,129,144,150,152,153,1,6,42,170,264,305,
%U 320,326,328,329,1,6,56,310,542,645,686,701,707,709,710
%N 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).
%H Andrew Howroyd, <a href="/A144215/b144215.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)
%H Rebecca Neville, <a href="https://web.archive.org/web/20191029092609/http://gtn.kazlow.info:80/GTN54.pdf">Graphs whose vertices are forests with bounded degree</a>, Graph Theory Notes of New York, LIV (2008), 12-21. [Wayback Machine link]
%F Column k is Euler transform of column k of A144528. - _Andrew Howroyd_, Dec 18 2020
%e Triangle begins:
%e 1
%e 1 2
%e 1 2 3
%e 1 3 5 6
%e 1 3 7 9 10
%e 1 4 11 17 19 20
%e 1 4 15 28 34 36 37
%e ...
%e From _Andrew Howroyd_, Dec 18 2020: (Start)
%e Formatted as an array to show the full columns:
%e ========================================================
%e n\k | 0 1 2 3 4 5 6 7 8 9 10
%e -----+--------------------------------------------------
%e 1 | 1 1 1 1 1 1 1 1 1 1 1 ...
%e 2 | 1 2 2 2 2 2 2 2 2 2 2 ...
%e 3 | 1 2 3 3 3 3 3 3 3 3 3 ...
%e 4 | 1 3 5 6 6 6 6 6 6 6 6 ...
%e 5 | 1 3 7 9 10 10 10 10 10 10 10 ...
%e 6 | 1 4 11 17 19 20 20 20 20 20 20 ...
%e 7 | 1 4 15 28 34 36 37 37 37 37 37 ...
%e 8 | 1 5 22 52 67 73 75 76 76 76 76 ...
%e 9 | 1 5 30 90 129 144 150 152 153 153 153 ...
%e 10 | 1 6 42 170 264 305 320 326 328 329 329 ...
%e 11 | 1 6 56 310 542 645 686 701 707 709 710 ...
%e 12 | 1 7 77 600 1161 1431 1536 1577 1592 1598 1600 ...
%e (End)
%o (PARI) \\ Here V(n,k) gives column k of A144528.
%o EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o 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))}
%o 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)}
%o M(n, m=n)={Mat(vector(m, k, EulerT(V(n,k-1)[2..1+n])~))}
%o { my(T=M(12)); for(n=1, #T~, print(T[n, 1..n])) } \\ _Andrew Howroyd_, Dec 18 2020
%Y The final diagonal gives A005195.
%Y Column k=2 is A000041.
%Y Cf. A144528, A144529, A339788.
%K nonn,tabl
%O 1,3
%A _N. J. A. Sloane_, Dec 20 2008
%E Terms a(29) and beyond from _Andrew Howroyd_, Dec 18 2020
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