A108527
Number of labeled mobiles (cycle rooted trees) with n generators.
Original entry on oeis.org
1, 3, 20, 229, 3764, 80383, 2107412, 65436033, 2347211812, 95492023811, 4344109422388, 218499395486909, 12039757564700644, 721239945304498215, 46669064731537444820, 3243864647191662324601, 241046155271316751794596
Offset: 1
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nmax=20; c[0]=0; A[x_]:=Sum[c[k]*x^k/k!,{k,0,nmax}]; Array[c,nmax]/.Solve[Rest[CoefficientList[Series[x-1-Log[1-A[x]]-(2-x)*A[x],{x,0,nmax}],x]]==0][[1]] (* Vaclav Kotesovec, Mar 28 2014 *)
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{a(n)=local(A=x+O(x^n)); for(i=0, n, A=intformal((1-A^2)/(1-x-2*A+x*A)+O(x^n))); n!*polcoeff(A, n)}
for(n=1, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Mar 28 2014
Original entry on oeis.org
1, 2, 9, 71, 800, 11659, 208173, 4398148, 107293711, 2967800711, 91777098006, 3137581240925, 117499040544197, 4783424590188490, 210333509575901445, 9934472399437068811, 501615620424564184408, 26963169913347131361647
Offset: 0
E.g.f.: A(x) = 1 + 1*x + 2*x^2/2! + 9*x^3/3! + 71*x^4/4! + ... =
exp(x + x^2/2! + 5*x^3/3! + 41*x^4/4! +... + A032188(n)*x^n/n! +...).
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Table[Sum[n!/(n-k)! * SeriesCoefficient[(x/(2*x + Log[1-x]))^(n + 1), {x, 0, k}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 01 2025 *)
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{a(n)=local(x=X+X^2*O(X^n));sum(k=0,n,n!/(n-k)!*polcoeff((x/(2*x+log(1-x)))^(n+1),k,X))}
A060694
A triangle related to rooted trees.
Original entry on oeis.org
1, 2, 1, 10, 8, 2, 82, 86, 36, 6, 938, 1202, 668, 192, 24, 13778, 20772, 14118, 5452, 1200, 120, 247210, 427828, 341122, 161688, 48312, 8640, 720, 5240338, 10228458, 9325398, 5184902, 1909920, 467784, 70560, 5040, 128149802, 278346286
Offset: 1
{1}, {2,1}, {10,8,2}, {82,86,36,6}, {938,1202,668,192,24}
A118787
Triangle where T(n,k) = n!*[x^k] ( x/(2*x + log(1-x)) )^(n+1), for n>=k>=0, read by rows.
Original entry on oeis.org
1, 1, 1, 2, 3, 5, 6, 12, 23, 41, 24, 60, 130, 255, 469, 120, 360, 870, 1860, 3679, 6889, 720, 2520, 6720, 15540, 32858, 65247, 123605, 5040, 20160, 58800, 146160, 328734, 689388, 1371887, 2620169, 40320, 181440, 574560, 1527120, 3638376, 8029980
Offset: 0
Triangle begins:
1;
1, 1;
2, 3, 5;
6, 12, 23, 41;
24, 60, 130, 255, 469;
120, 360, 870, 1860, 3679, 6889;
720, 2520, 6720, 15540, 32858, 65247, 123605;
5040, 20160, 58800, 146160, 328734, 689388, 1371887, 2620169; ...
Triangle is formed from powers of F(x) = x/(2*x + log(1-x)):
F(x)^1 = (1) + 1/2*x + 7/12*x^2 + 17/24*x^3 + 629/720*x^4 +...
F(x)^2 = (1 + x)/1! +17/12*x^2 + 2*x^3 + 671/240*x^4 ...
F(x)^3 = (2 + 3*x + 5*x^2)/2! + 4*x^3 + 1489/240*x^4 +...
F(x)^4 = (6 + 12*x + 23*x^2 + 41/6*x^3)/3! + 8351/720*x^4 +...
F(x)^5 = (24 + 60*x + 130*x^2 + 255*x^3 + 469*x^4)/4! +...
A269957
Triangle read by rows, T(n,k) = Sum_{j=k..n} A269940(n,j)*A269939(j,k), for n>=0 and 0<=k<=n.
Original entry on oeis.org
1, 0, 1, 0, 5, 9, 0, 41, 210, 225, 0, 469, 5115, 14175, 11025, 0, 6889, 142492, 763350, 1455300, 893025, 0, 123605, 4566149, 41943090, 146522250, 212837625, 108056025, 0, 2620169, 166939742, 2462128095, 13973628900, 35936814375, 42141849750, 18261468225
Offset: 0
Triangle starts:
[1]
[0, 1]
[0, 5, 9]
[0, 41, 210, 225]
[0, 469, 5115, 14175, 11025]
[0, 6889, 142492, 763350, 1455300, 893025]
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F = lambda n,k,f: sum((-1)^(m+k)*binomial(n+k,n+m)*f(n+m,m) for m in (0..k))
T = lambda n,k: sum(F(n, j, stirling_number1)*F(j, k, stirling_number2) for j in (k..n))
for n in (0..6): print([T(n, k) for k in (0..n)])
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