A000949
Number of forests with n nodes and height at most 2.
Original entry on oeis.org
1, 1, 3, 16, 101, 756, 6607, 65794, 733833, 9046648, 121961051, 1782690174, 28055070397, 472594822324, 8479144213191, 161340195463066, 3243707386310033, 68679247688467056, 1526976223741111987, 35557878951515668726, 865217354118762606021
Offset: 0
G.f. = 1 + x + 3*x^2 + 16*x^3 + 101*x^4 + 756*x^5 + 6607*x^6 + 65794*x^7 + ... - _Michael Somos_, Jul 03 2018
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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nn = 20; Range[0, nn]! CoefficientList[Series[Exp[x*Exp[x*Exp[x]]], {x, 0, nn}], x] (* T. D. Noe, Jun 21 2012 *)
a[ n_] := If[ n < 0, 0, 1 + n! Sum[ Sum[ k^(n - m - k) m^k / (k! (n - m - k)!), {k, n - m}] / m!, {m, n - 1}]]; (* Michael Somos, Jul 03 2018 *)
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a(n):=n!*sum(sum((k^(n-m-k)*m^k)/(k!*(n-m-k)!),k,1,n-m)/m!,m,1,n-1)+1; /* Vladimir Kruchinin, May 28 2011 */
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x='x+O('x^66); Vec(serlaplace(exp(x*exp(x*exp(x))))) /* show terms with a(0)=1 */ /* Joerg Arndt, May 28 2011 */
A000951
Number of forests with n nodes and height at most 4.
Original entry on oeis.org
1, 3, 16, 125, 1296, 16087, 229384, 3687609, 66025360, 1303751051, 28151798544, 659841763957, 16681231615816, 452357366282655, 13095632549137576, 403040561722348913, 13138626717852194976, 452179922268565180819, 16381932383826669204640
Offset: 1
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
-
nn = 20; Range[0, nn]! CoefficientList[Series[Exp[x*Exp[x*Exp[x*Exp[x*Exp[x]]]]], {x, 0, nn}], x] (* T. D. Noe, Jun 21 2012 *)
A000950
Number of forests with n nodes and height at most 3.
Original entry on oeis.org
1, 3, 16, 125, 1176, 12847, 160504, 2261289, 35464816, 612419291, 11539360944, 235469524237, 5170808565976, 121535533284999, 3043254281853496, 80852247370051793, 2270951670959226336, 67221368736302224819, 2091039845329887687136
Offset: 1
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
-
nn = 20; Range[0, nn]! CoefficientList[Series[Exp[x*Exp[x*Exp[x*Exp[x]]]], {x, 0, nn}], x] (* T. D. Noe, Jun 21 2012 *)
A052513
Number of labeled trees of height at most 3.
Original entry on oeis.org
0, 1, 2, 9, 64, 505, 4536, 46249, 526352, 6604497, 90466480, 1341571561, 21392282088, 364715915161, 6616327512536, 127187163197865, 2581443127409056, 55143025567270561, 1236226458392407008, 29012548251081127753, 711157579030313374520, 18169564436494014726441
Offset: 0
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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m:=20; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( x*Exp(x*Exp(x*Exp(x))) )); [0] cat [Factorial(n)*b[n]: n in [1..m-1]]; // G. C. Greubel, May 13 2019
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spec := [S,{S=Prod(Z,Set(T1)), T2=Prod(Z,Set(T3)), T3=Z, T1=Prod(Z,Set(T2))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
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nn=20;a=x Exp[x];b=x Exp[a];Range[0,nn]! CoefficientList[Series[x Exp[b],{x,0,nn}],x] (* Geoffrey Critzer, Sep 20 2012 *)
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N=33; x='x+O('x^N);
egf=x*exp(x*exp(x*exp(x)));
v=Vec(serlaplace(egf));
vector(#v+1,n,if(n==1,0,v[n-1]))
/* Joerg Arndt, Sep 15 2012 */
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m = 20; T = taylor(x*exp(x*exp(x*exp(x))), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 13 2019
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