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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). 5
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

R. Neville, Graphs whose vertices are forests with bounded degree, Graph Theory Notes of New York, LIV (2008), 12-21.

FORMULA

Column k is Euler transform of column k of A144528. - Andrew Howroyd, Dec 18 2020

EXAMPLE

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)

PROG

(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

CROSSREFS

The final diagonal gives A005195.

Column k=2 is A000041.

Cf. A144528, A144529, A339788.

Sequence in context: A104468 A293003 A110062 * A254539 A283827 A122087

Adjacent sequences: A144212 A144213 A144214 * A144216 A144217 A144218

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Dec 20 2008

EXTENSIONS

Terms a(29) and beyond from Andrew Howroyd, Dec 18 2020

STATUS

approved

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Last modified December 2 11:37 EST 2022. Contains 358493 sequences. (Running on oeis4.)