A143543 Triangle read by rows: T(n,k) = number of labeled graphs on n nodes with k connected components, 1<=k<=n.
1, 1, 1, 4, 3, 1, 38, 19, 6, 1, 728, 230, 55, 10, 1, 26704, 5098, 825, 125, 15, 1, 1866256, 207536, 20818, 2275, 245, 21, 1, 251548592, 15891372, 925036, 64673, 5320, 434, 28, 1, 66296291072, 2343580752, 76321756, 3102204, 169113, 11088, 714, 36, 1
Offset: 1
Examples
The triangle T(n,k) starts as: n=1: 1; n=2: 1, 1; n=3: 4, 3, 1; n=4: 38, 19, 6, 1; n=5: 728, 230, 55, 10, 1; n=6: 26704, 5098, 825, 125, 15, 1; ...
Links
- Alois P. Heinz, Rows n = 1..82, flattened (first 25 rows from Marko Riedel)
- Marko Riedel, Maple implementation of memoized recurrence.
- Marko Riedel et al., Proof of recurrence relation.
Crossrefs
Programs
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Maple
g:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-add( binomial(n, k)*2^((n-k)*(n-k-1)/2)*g(k)*k, k=1..n-1)/n) end: b:= proc(n) option remember; `if`(n=0, 1, add(expand( b(n-j)*binomial(n-1, j-1)*g(j)*x), j=1..n)) end: T:= (n, k)-> coeff(b(n$2), x, k): seq(seq(T(n, k), k=1..n), n=1..10); # Alois P. Heinz, Feb 02 2024 -
Mathematica
a= Sum[2^Binomial[n,2] x^n/n!,{n,0,10}]; Rest[Transpose[Table[Range[0, 10]! CoefficientList[Series[Log[a]^n/n!, {x, 0, 10}], x], {n, 1, 10}]]] // Grid (* Geoffrey Critzer, Mar 15 2011 *) -
PARI
T(n)={[Vecrev(p/y) | p <- Vec(serlaplace(exp(y*log(sum(k=0, n, 2^binomial(k,2)*x^k/k!, O(x*x^n))))))]} { foreach(T(8), row, print(row)) } \\ Andrew Howroyd, Jun 14 2025 -
Sage
# uses[bell_matrix from A264428, A001187] # Adds a column 1,0,0,0, ... at the left side of the triangle. bell_matrix(lambda n: A001187(n+1), 9) # Peter Luschny, Jan 17 2016
Formula
SUM[n,k=0..oo] T(n,k) * x^n * y^k / n! = exp( y*( F(x) - 1 ) ) = ( SUM[n=0..oo] 2^binomial(n, 2)*x^n/n! )^y, where F(x) is e.g.f. of A001187.
T(n,k) = Sum_{q=0..n-1} C(n-1, q) T(q, k-1) 2^C(n-q,2) - Sum_{q=0..n-2} C(n-1, q) T(q+1, k) 2^C(n-1-q, 2) where T(0,0) = 1 and T(0,k) = 0 and T(n,0) = 0. - Marko Riedel, Feb 04 2019
Sum_{k=1..n} k * T(n,k) = A125207(n) - Alois P. Heinz, Feb 02 2024
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