A029856
Number of rooted trees with 2-colored leaves.
Original entry on oeis.org
2, 2, 5, 13, 37, 108, 332, 1042, 3360, 11019, 36722, 123875, 422449, 1453553, 5040816, 17599468, 61814275, 218252584, 774226549, 2758043727, 9862357697, 35387662266, 127374191687, 459783039109, 1664042970924, 6037070913558, 21951214425140, 79981665585029
Offset: 1
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A:= proc(n) option remember; if n=0 then 0 else convert(series(x+x* exp(sum(subs(x=x^i, A(n-1))/i, i=1..n-1)), x=0, n+1), polynom) fi end; a:= n-> coeff(A(n), x,n): seq(a(n), n=1..25); # Alois P. Heinz, Aug 22 2008
# second Maple program:
with(numtheory): a:= proc(n) option remember; local d,j; if n<=1 then 2*n else (add(d*a(d), d=divisors(n-1)) +add(add(d*a(d), d=divisors(j)) *a(n-j), j=1..n-2))/ (n-1) fi end: seq(a(n), n=1..25); # Alois P. Heinz, Sep 06 2008
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a[n_] := a[n] = If [n <= 1, 2*n, (Sum[d*a[d], {d, Divisors[n-1]}] + Sum[Sum[d*a[d], {d, Divisors[j]}]*a[n-j], {j, 1, n-2}])/(n-1)]; Array[a, 25] (* Jean-François Alcover, Mar 13 2015, after Alois P. Heinz *)
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{a(n)=local(A=x+x*O(x^n));for(i=1,n, A=x+x*exp(sum(m=1,n,subst(A,x,x^m)/m)));polcoeff(A,n,x)} \\ Paul D. Hanna, Oct 19 2005
A362389
G.f. satisfies A(x) = exp( Sum_{k>=1} (2^k + A(x^k)) * x^k/k ).
Original entry on oeis.org
1, 3, 10, 34, 122, 450, 1723, 6758, 27135, 110913, 460395, 1935233, 8222504, 35255000, 152353021, 662892684, 2901595559, 12768195617, 56450822365, 250637657015, 1117060889815, 4995815027658, 22413020866875, 100842092305575, 454912716037387
Offset: 0
-
seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, (2^k+subst(A, x, x^k))*x^k/k)+x*O(x^n))); Vec(A);
A363507
G.f. satisfies A(x) = exp( Sum_{k>=1} (3 + A(x^k)) * x^k/k ).
Original entry on oeis.org
1, 4, 14, 50, 191, 763, 3180, 13640, 59937, 268304, 1219626, 5614038, 26117296, 122598622, 579977691, 2762264225, 13234003724, 63737225733, 308406648979, 1498558628584, 7309116199687, 35772044402485, 175621484712091, 864670723348447
Offset: 0
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seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, (3+subst(A, x, x^k))*x^k/k)+x*O(x^n))); Vec(A);
A038050
Number of labeled rooted trees with 3-colored leaves.
Original entry on oeis.org
3, 6, 45, 504, 7785, 153468, 3681909, 104126256, 3392064945, 125089571700, 5151335388309, 234322765501608, 11668410187187481, 631335472193760012, 36881146426978035765, 2313552152470193124192, 155107536736245864549345
Offset: 1
- F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 185 (3.1.83)
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Rest[CoefficientList[Series[2*x-LambertW[-x*E^(2*x)], {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Oct 05 2013 *)
A363508
G.f. satisfies A(x) = exp( Sum_{k>=1} (4 + A(x^k)) * x^k/k ).
Original entry on oeis.org
1, 5, 20, 80, 340, 1516, 7046, 33736, 165436, 826566, 4193348, 21542664, 111848161, 585949358, 3093526496, 16442687695, 87914559018, 472522551440, 2551591234444, 13836226412386, 75311992329508, 411336641019998, 2253641429297336
Offset: 0
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seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, (4+subst(A, x, x^k))*x^k/k)+x*O(x^n))); Vec(A);
A363547
G.f. satisfies A(x) = exp( Sum_{k>=1} A(x^k) * x^k/(k * (1 - x^k)^2) ).
Original entry on oeis.org
1, 1, 4, 13, 47, 168, 635, 2420, 9460, 37445, 150309, 609568, 2495710, 10298332, 42793974, 178910161, 752034697, 3176346092, 13473881397, 57378127986, 245205968960, 1051257068207, 4520229295852, 19488595397346, 84231899582543, 364893870958302
Offset: 0
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seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, subst(A, x, x^k)*x^k/(k*(1-x^k)^2))+x*O(x^n))); Vec(A);
A036252
Number of trees with 3-colored leaves.
Original entry on oeis.org
1, 3, 6, 6, 16, 39, 114, 335, 1081, 3574, 12408, 44076, 160915, 598244, 2263400, 8681464, 33713947, 132305267, 524095596, 2093208435, 8422013745, 34110403728, 138979989162, 569339728312, 2343898451275, 9693334574919
Offset: 0
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