A064781
Number of partially labeled rooted trees with n nodes (5 of which are labeled).
Original entry on oeis.org
625, 7776, 60387, 373895, 2019348, 9941905, 45765687, 200160586, 840783867, 3418136039, 13525307849, 52314181361, 198451152867, 740283848220, 2721325964209, 9875523648986, 35429691205221, 125814678386528
Offset: 5
- J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 134.
A064782
Number of partially labeled rooted trees with n nodes (6 of which are labeled).
Original entry on oeis.org
7776, 117649, 1071904, 7601777, 46205469, 252620693, 1277843841, 6088987597, 27669884619, 120970585689, 512143134419, 2110092318654, 8493720340735, 33506301271642, 129860760881905, 495506657571553, 1864611086321279
Offset: 6
- J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 134.
A064785
Number of partially labeled trees with n nodes (6 of which are labeled).
Original entry on oeis.org
1296, 16807, 134960, 858578, 4741835, 23786827, 111254536, 493289047, 2096891419, 8614217489, 34402073301, 134162057607, 512703873915, 1925300176534, 7120276125066, 25981116938906, 93678940211218
Offset: 6
- J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 138.
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b:= proc(n) option remember; if n<=1 then n else add(k*b(k)* s(n-1, k), k=1..n-1)/(n-1) fi end: s:= proc(n,k) option remember; add(b(n+1-j*k), j=1..iquo(n,k)) end: B:= proc(n) option remember; add(b(k)*x^k, k=1..n) end: a:= n-> coeff(series(B(n-2)^6* (1296-2633* B(n-2)+ 2128*B(n-2)^2 -806*B(n-2)^3 +120*B(n-2)^4)/ (1-B(n-2))^9, x=0, n+1),x,n): seq(a(n), n=6..22); # Alois P. Heinz, Aug 22 2008
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jmax = 23; B[_] = 0;
Do[B[x_] = x*Exp[Sum[B[x^k]/k, {k, 1, j}]]+O[x]^j // Normal, {j, 1, jmax}];
A[x_] = B[x]^6*(1296 - 2633*B[x] + 2128*B[x]^2 - 806*B[x]^3 + 120*B[x]^4)/ (1 - B[x])^9;
CoefficientList[A[x] + O[x]^jmax, x] // Drop[#, 6]& (* Jean-François Alcover, Apr 25 2022 *)