A108521
Number of rooted trees with n generators.
Original entry on oeis.org
1, 2, 5, 16, 53, 194, 730, 2868, 11526, 47370, 197786, 837467, 3585696, 15501423, 67563442, 296579626, 1309973823, 5817855174, 25964218471, 116379947718, 523699384013, 2364967753113, 10714396241046, 48684193997623
Offset: 1
-
a[1] = 1; a[n_] := a[n] = 1+a[n-1]+Total[Product[Binomial[a[i]-1+Count[#,i], Count[#,i]], {i, DeleteCases[DeleteDuplicates[#],1]}]&/@ IntegerPartitions[n,{2,n-1}]]; Table[a[n],{n,24}] (* Robert A. Russell, Jun 02 2020 *)
a[1] = 1; a[n_] := a[n] = a[n-1] + (DivisorSum[n, a[#] # &, #Robert A. Russell, Jun 04 2020 *)
-
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=[1]); for(n=2, n, v=concat(v, v[#v] + EulerT(concat(v,[0]))[n])); v} \\ Andrew Howroyd, Aug 31 2018
A108522
Number of increasing rooted trees with n generators.
Original entry on oeis.org
1, 2, 9, 70, 771, 10948, 190205, 3907494, 92654059, 2490459468, 74827519077, 2485153213814, 90403692195179, 3574835773247140, 152675377606343901, 7003761877546096278, 343454890456254782203, 17929588055863943650988
Offset: 1
-
{a(n)=local(A=x);for(i=1,n,A=intformal((1+A)/(2-exp(A+x*O(x^n)))) );n!*polcoeff(A,n)}
for(n=1,20,print1(a(n),", ")) \\ Paul D. Hanna, Mar 29 2014
A007152
Evolutionary trees of magnitude n.
Original entry on oeis.org
1, 1, 4, 28, 301, 4466, 84974, 1974904, 54233540, 1718280152, 61695193880, 2475688513024, 109797950475448, 5333253012414224, 281576039542538368, 16055279332196218624, 983264280857581866112, 64369946360185677026048, 4485859513184032011682304, 331558482325457407154881024
Offset: 1
- L. R. Foulds and R. W. Robinson, Counting certain classes of evolutionary trees with singleton labels, Congress. Num., 44 (1984), 65-88.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
-
Q := proc(n)
option remember ;
if n <= 1 then
0;
else
A007151(n)-A007151(n-1) +(n-1)*procname(n-1) ; # eq (3.5)
%/2 ;
end if;
end proc:
A007152 := proc(n)
if n = 1 then
1;
else
A007151(n-1)+Q(n-1) ; # eq (3.9)
end if ;
end proc:
seq(A007152(n),n=1..20 ); # R. J. Mathar, Mar 19 2018
-
m = 20;
A007151 = Rest[Range[0, m]! CoefficientList[ InverseSeries[ Series[(2x - E^x + 1)/(x + 1), {x, 0, m}], x], x]];
Q[n_] := Q[n] = If[n <= 1, 0, (1/2)(-A007151[[n - 1]] + A007151[[n]] + (n - 1) Q[n - 1])];
a[n_] := If[n == 1, 1, A007151[[n - 1]] + Q[n - 1]];
Array[a, m] (* Jean-François Alcover, Mar 30 2020, from Maple *)
Showing 1-3 of 3 results.
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