A244531 Number of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 2.
1, 0, 2, 5, 11, 28, 78, 201, 532, 1441, 3895, 10569, 28926, 79493, 219226, 607189, 1687880, 4706737, 13165215, 36929595, 103860429, 292808814, 827392709, 2342964435, 6647953886, 18898472568, 53818654942, 153518738980, 438602656951, 1254943919799, 3595714927194
Offset: 3
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 3..1000
Programs
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Maple
b:= proc(n, t, k) option remember; `if`(n=0, `if`(t in [0, k], 1, 0), `if`(t>n, 0, add(b(j-1, k$2)* b(n-j, max(0, t-1), k), j=1..n))) end: a:= n-> b(n-1, 2$2) -b(n-1, 3$2): seq(a(n), n=3..50); -
Mathematica
b[n_, t_, k_] := b[n, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[t > n, 0, Sum[b[j - 1, k, k]*b[n - j, Max[0, t - 1], k], {j, 1, n}]]]; T[n_, k_] := b[n - 1, k, k] - If[n == 1 && k == 0, 0, b[n - 1, k + 1, k + 1]]; a[n_] := b[n - 1, 2, 2] - b[n - 1, 3, 3]; Table[a[n], {n, 3, 50}] (* Jean-François Alcover, Feb 06 2015, after Maple *)
Formula
Recurrence: (n-2)*n*(n+1)*(31556*n^6 - 602602*n^5 + 4562565*n^4 - 17272550*n^3 + 33523297*n^2 - 29665770*n + 7578864)*a(n) = -2*(n-4)*n*(15778*n^6 - 93541*n^5 - 718683*n^4 + 7746097*n^3 - 25426183*n^2 + 35870760*n - 18623988)*a(n-1) + 2*(189336*n^9 - 4357178*n^8 + 42198478*n^7 - 222932639*n^6 + 692179375*n^5 - 1246825745*n^4 + 1121148607*n^3 - 95771898*n^2 - 622360656*n + 342066240)*a(n-2) + 4*(15778*n^9 - 301301*n^8 + 2556736*n^7 - 13524389*n^6 + 51959635*n^5 - 145042550*n^4 + 255185823*n^3 - 199177680*n^2 - 62590212*n + 146335680)*a(n-3) - 2*(n-4)*(63112*n^8 - 1252538*n^7 + 9554713*n^6 - 31464554*n^5 + 11620330*n^4 + 221568106*n^3 - 627283143*n^2 + 624591414*n - 146644560)*a(n-4) - 4*(n-5)*(n-4)*(504896*n^7 - 9428629*n^6 + 69275668*n^5 - 250040744*n^4 + 437755491*n^3 - 253595994*n^2 - 179277570*n + 187109352)*a(n-5) - 69*(n-6)*(n-5)*(n-4)*(31556*n^6 - 413266*n^5 + 2022895*n^4 - 4417190*n^3 + 3528357*n^2 + 989760*n - 1844640)*a(n-6). - Vaclav Kotesovec, Jul 02 2014
a(n) ~ 3^(n+1/2) / (8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jul 02 2014