A108643 -id:A108643 - cp's OEIS frontend

A095830 Number of binary trees of path length n.

1, 2, 1, 4, 4, 2, 14, 8, 12, 28, 21, 52, 52, 72, 92, 160, 212, 178, 446, 360, 628, 920, 918, 1568, 1784, 2676, 2960, 4724, 5360, 7280, 10876, 10936, 17484, 21732, 28469, 34224, 48648, 61232, 78196, 105680, 120904, 178848, 217404, 279312

Offset: 0

Gadiel Seroussi (seroussi(AT)hpl.hp.com), Jul 10 2004

nonn

The cited preprint gives an asymptotic estimate for the number of trees as the path length goes to infinity, for t-ary trees, t >= 2. This sequence corresponds to t=2.

			a(1) = 2 because there are two binary trees of path length 1: a root with a left child and a root with a right child.
a(2) = 1 because there is just one binary tree of path length 2: a root with its two children.

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 0..200 from Vincenzo Librandi)
G. Seroussi, On the number of t-ary trees with a given path length, arXiv:cs/0509046 [cs.DM], 2005-2007; Algorithmica 46(3), 557-565, 2006.

Cf. A106182.

terms = 44; B[, ] = 0;
Do[B[w_, z_] = Series[z B[w, w z]^2 + 1, {w, 0, terms-1}, {z, 0, terms-1}] // Normal, {terms-1}];
CoefficientList[B[w, 1] - 1, w] (* Jean-François Alcover, Dec 03 2018 *)

G.f.: B(w, 1) - 1, where B(w, z) satisfies the functional equation B(w, z) = z B(w, wz)^2 + 1. B(w, z) is the g.f. for the number of binary trees of given path length and number of nodes (see Knuth Vol. 1 Sec. 2.3.4.5); B(1, z) is the g.f. for the Catalan numbers; for B(w, w) see A108643.

A132330 G.f.: A(x) = 1 + x(A_2)^3; A_2 = 1 + x^2(A_3)^3; A_3 = 1 + x^3(A_4)^3; ... A_n = 1 + x^n(A_{n+1})^3 for n>=1 with A_1 = A(x).

Original entry on oeis.org

1, 1, 0, 3, 0, 3, 9, 1, 18, 9, 36, 45, 57, 90, 114, 351, 165, 558, 738, 1044, 1791, 1908, 3915, 4926, 8568, 8553, 17217, 26271, 30474, 50967, 68526, 113319, 144324, 219195, 299359, 473454, 665424, 860733, 1396350, 1895913, 2762550, 3790935, 5695974

Offset: 0

Paul D. Hanna, Aug 20 2007

nonn

Cf. A132331 (cube); A001764; A108643 (variant).

{a(n)=local(A=1+x*O(x^n)); for(j=0,n-1,A=1+x^(n-j)*A^3);polcoeff(A,n)}

G.f. A(x) = B(x,x), where B(w,x) satisfies the functional equation B(w,x) = 1 + x*B(w,wx)^3. B(w,x) is the g.f. for the number of ternary trees of given path length and number of nodes; B(1,x) is the g.f. for A001764.

A228866 G.f.: A(x) = 1 + xB(x), where B(x) = 1 + x^2C(x)^2, C(x) = 1 + x^3D(x)^3, D(x) = 1 + x^4E(x)^4, ...

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 2, 0, 0, 1, 6, 0, 0, 6, 6, 24, 0, 15, 26, 48, 36, 140, 120, 60, 288, 279, 600, 660, 1476, 822, 2166, 2880, 5100, 7047, 6300, 14100, 21440, 30210, 30054, 62496, 72060, 123180, 174780, 253980, 319488, 497544, 730560, 976020, 1654856, 1997706, 3085932, 4160740, 6426480

Offset: 0

Paul D. Hanna, Sep 06 2013

nonn

			G.f.: A(x) = 1 + x + x^3 + 2*x^6 + x^9 + 6*x^10 + 6*x^13 + 6*x^14 + 24*x^15 +...
where A(x) = 1 + x*B(x),
B(x) = 1 + x^2 + 2*x^5 + x^8 + 6*x^9 + 6*x^12 + 6*x^13 + 24*x^14 +...
B(x) = 1 + x^2*C(x)^2,
C(x) = 1 + x^3 + 3*x^7 + 3*x^11 + 12*x^12 + x^15 + 24*x^16 + 18*x^17 +...
C(x) = 1 + x^3*D(x)^3,
D(x) = 1 + x^4 + 4*x^9 + 6*x^14 + 20*x^15 + 4*x^19 + 60*x^20 + 40*x^21 +...
D(x) = 1 + x^4*E(x)^4,
E(x) = 1 + x^5 + 5*x^11 + 10*x^17 + 30*x^18 + 10*x^23 + 120*x^24 + 75*x^25 +...
E(x) = 1 + x^5*F(x)^5,
F(x) = 1 + x^6 + 6*x^13 + 15*x^20 + 42*x^21 + 20*x^27 + 210*x^28 + 126*x^29 +...
F(x) = 1 + x^6*G(x)^6,
G(x) = 1 + x^7 + 7*x^15 + 21*x^23 + 56*x^24 + 35*x^31 + 336*x^32 + 196*x^33 +...
G(x) = 1 + x^7*H(x)^7,
H(x) = 1 + x^8 + 8*x^17 + 28*x^26 + 72*x^27 + 56*x^35 + 504*x^36 + 288*x^37 +...
...

Paul D. Hanna, Table of n, a(n) for n = 0..500

Cf. A108643, A095793.

{a(n)=local(A=1);for(k=1,n,A = 1 + (x*A)^(n-k+1) +x*O(x^n));polcoeff(A,n)}
for(n=0,120,print1(a(n),", "))

A302693 a(n) = [x^n] 1 + x(1 + x^2(1 + x^3(1 + x^4(1 + x^5*(1 + ...)^n)^n)^n)^n)^n.

Original entry on oeis.org

1, 1, 0, 3, 0, 10, 36, 35, 448, 450, 4600, 13167, 45408, 240006, 670124, 3863835, 13209088, 60533974, 268324056, 1089776654, 5328589520, 22065836226, 106289169448, 481033496803, 2231768294208, 10748702587800, 49858509230920, 245664498591891, 1175943617623264

Offset: 0

Ilya Gutkovskiy, Apr 11 2018

nonn

Cf. A010054, A108643, A132330, A302657, A302686.

Table[SeriesCoefficient[1 + x Fold[(x^(#2 + 1) #1 + 1)^n &, 0, Reverse[Range[n]]], {x, 0, n}], {n, 0, 28}]

A302751 Expansion of 1 + x(1 + 2x^2(1 + 3x^3(1 + 4x^4*(1 + ...)^4)^3)^2).

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 12, 0, 0, 18, 144, 0, 0, 432, 576, 2880, 0, 4320, 9408, 23040, 21600, 109440, 172800, 110880, 662400, 832320, 2678400, 4060800, 10296000, 9412992, 32922000, 63676800, 135734400, 263556528, 281030400, 973036800, 1906704000, 4069224000, 5184984960

Offset: 0

Ilya Gutkovskiy, Apr 12 2018

nonn

			G.f. A(x) = 1 + x + 2*x^3 + 12*x^6 + 18*x^9 + 144*x^10 + 432*x^13 + 576*x^14 + 2880*x^15 + ...

Cf. A095793, A108643, A128318, A228866, A302688.

nmax = 38; CoefficientList[Series[1 + x Fold[((#2 + 1) x^(#2 + 1) #1 + 1)^#2 &, 0, Reverse[Range[nmax]]], {x, 0, nmax}], x]

cp's OEIS Frontend

A095830 Number of binary trees of path length n.

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A132330 G.f.: A(x) = 1 + x*(A_2)^3; A_2 = 1 + x^2*(A_3)^3; A_3 = 1 + x^3*(A_4)^3; ... A_n = 1 + x^n*(A_{n+1})^3 for n>=1 with A_1 = A(x).

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A228866 G.f.: A(x) = 1 + x*B(x), where B(x) = 1 + x^2*C(x)^2, C(x) = 1 + x^3*D(x)^3, D(x) = 1 + x^4*E(x)^4, ...

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A302693 a(n) = [x^n] 1 + x*(1 + x^2*(1 + x^3*(1 + x^4*(1 + x^5*(1 + ...)^n)^n)^n)^n)^n.

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A302751 Expansion of 1 + x*(1 + 2*x^2*(1 + 3*x^3*(1 + 4*x^4*(1 + ...)^4)^3)^2).

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A132330 G.f.: A(x) = 1 + x(A_2)^3; A_2 = 1 + x^2(A_3)^3; A_3 = 1 + x^3(A_4)^3; ... A_n = 1 + x^n(A_{n+1})^3 for n>=1 with A_1 = A(x).

A228866 G.f.: A(x) = 1 + xB(x), where B(x) = 1 + x^2C(x)^2, C(x) = 1 + x^3D(x)^3, D(x) = 1 + x^4E(x)^4, ...

A302693 a(n) = [x^n] 1 + x(1 + x^2(1 + x^3(1 + x^4(1 + x^5*(1 + ...)^n)^n)^n)^n)^n.

A302751 Expansion of 1 + x(1 + 2x^2(1 + 3x^3(1 + 4x^4*(1 + ...)^4)^3)^2).