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 A036650 Number of 5-valent trees with n nodes. 4
 1, 1, 1, 1, 2, 3, 6, 10, 21, 42, 94, 204, 473, 1098, 2633, 6353, 15641, 38789, 97416, 246410, 628726, 1614292, 4171955, 10839366, 28308678, 74266477, 195667533, 517504253, 1373640355, 3658205088, 9772510063, 26181295237, 70330621171 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS Andrew Howroyd, Table of n, a(n) for n = 0..500 R. Otter, The number of trees, Ann. of Math. (2) 49 (1948), 583-599 discusses asymptotics. E. M. Rains and N. J. A. Sloane, On Cayley's Enumeration of Alkanes (or 4-Valent Trees)., J. Integer Sequences, Vol. 2 (1999), Article 99.1.1. FORMULA a(n) = A036648(n) + A036649(n) for n > 0. MATHEMATICA n = 30; (* algorithm from Rains and Sloane *) S4[f_, h_, x_] := f[h, x]^4/24 + f[h, x]^2 f[h, x^2]/4 + f[h, x] f[h, x^3]/3 + f[h, x^2]^2/8 + f[h, x^4]/4; S5[f_, h_, x_] := f[h, x]^5/120 + f[h, x]^3 f[h, x^2]/12 + f[h, x]^2 f[h, x^3]/6 + f[h, x] f[h, x^2]^2/8 + f[h, x] f[h, x^4]/4 + f[h, x^2] f[h, x^3]/6 + f[h, x^5]/5; T[-1, z_] := 1;  T[h_, z_] := T[h, z] = Table[z^k, {k, 0, n}].Take[CoefficientList[z^(n+1) + 1 + S4[T, h-1, z]z, z], n+1]; Sum[Take[CoefficientList[z^(n+1) + S5[T, h-1, z]z - S5[T, h-2, z]z - (T[h-1, z] - T[h-2, z]) (T[h-1, z]-1), z], n+1], {h, 1, n/2}] + PadRight[{0, 1}, n+1] + Sum[Take[CoefficientList[z^(n+1) + (T[h, z] - T[h-1, z])^2/2 + (T[h, z^2] - T[h-1, z^2])/2, z], n+1], {h, 0, n/2}] (* Robert A. Russell, Sep 15 2018 *) CROSSREFS Column k=5 of A144528. Cf. A036648, A036649. Sequence in context: A265582 A242563 A240513 * A049889 A014270 A127076 Adjacent sequences:  A036647 A036648 A036649 * A036651 A036652 A036653 KEYWORD nonn AUTHOR EXTENSIONS a(0) changed to 1 by Andrew Howroyd, Dec 18 2020 STATUS approved

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Last modified August 14 22:45 EDT 2022. Contains 356122 sequences. (Running on oeis4.)