A085748 -id:A085748

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A091190

G.f. A(x) satisfies x*A(x)^3 = B(x*A(x^3)) where B(x) = x/(1 - 3*x).

+10
5

1, 1, 2, 5, 13, 35, 97, 273, 778, 2240, 6499, 18976, 55703, 164243, 486130, 1443620, 4299365, 12836825, 38413933, 115184282, 346005073, 1041072108, 3137060983, 9465689545, 28596915843, 86492865522, 261876842801, 793661873276

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

More generally, given A(x) satisfies x*A(x)^p = B(x*A(x^p)) where B(x) = x/(1-p*x), then it appears that A(x) is an integer series only when p is prime. This is a special case of sequences with g.f.s that satisfy the more general functional equation x*A(x)^m = B(x*A(x^m)) studied by Michael Somos; some other examples are A085748, A091188 and A091200.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..900

FORMULA

From Paul D. Hanna, Mar 09 2024: (Start)

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

(1) A(x)^3 = A(x^3) / (1 - 3*x*A(x^3)).

(2) A(x) = x/Series_Reversion(D(x)) where D(x) = x*A(D(x)) is the g.f. of A370441.

(End)

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 35*x^5 + 97*x^6 + 273*x^7 + 778*x^8 + 2240*x^9 + 6499*x^10 + 18976*x^11 + 55703*x^12 + ...

where A(x)^3 = A(x^3) / (1 - 3*x*A(x^3)).

RELATED SERIES.

A(x)^3 = 1 + 3*x + 9*x^2 + 28*x^3 + 87*x^4 + 270*x^5 + 839*x^6 + 2607*x^7 + 8100*x^8 + 25169*x^9 + 78207*x^10 + 243009*x^11 + 755095*x^12 + ...

Also, D(x) = x*A(D(x)) is the g.f. of A370441, which begins

D(x) = x + x^2 + 3*x^3 + 12*x^4 + 54*x^5 + 261*x^6 + 1324*x^7 + 6952*x^8 + 37461*x^9 + ... + A370441(n)*x^n + ...

such that D(x)^3 = D( x^3 + 3*D(x)^4 ).

MATHEMATICA

m = 28; B[x_] = x/(1 - 3 x); A[_] = 1;

Do[A[x_] = (B[x A[x^3]]/x)^(1/3) + O[x]^m // Normal, {m}];

CoefficientList[A[x], x] (* Jean-François Alcover, Oct 29 2019 *)

PROG

(PARI) {a(n) = my(A, p=3, m=1); if(n<0, 0, m=1; A=1+O(x); while(m<=n, m*=p; A = x*subst(A, x, x^p); A = (A/(1-p*A)/x)^(1/p)); polcoeff(A, n))}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A085748, A091188, A091200, A370441.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 22 2004

EXTENSIONS

Corrected by T. D. Noe, Oct 25 2006

STATUS

approved

A091200

G.f. A(x) satisfies xA(x)^5 = B(xA(x^5)) where B(x) = x/(1-5x).

+10
4

1, 1, 3, 11, 44, 185, 802, 3553, 15994, 72886, 335387, 1555487, 7261310, 34083382, 160730900, 761039051, 3616102911, 17235223345, 82372594183, 394648349447, 1894921311499, 9116598414141, 43939539520427, 212124129983285

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

More generally, given A(x) satisfies xA(x)^p = B(xA(x^p)) where B(x) = x/(1-p*x), then it appears that A(x) is an integer series only when p is prime. This is a special case of sequences with g.f.s that satisfy the more general functional equation xA(x)^m = B(xA(x^m)) originated by Michael Somos; some other examples are A085748, A091188 and A091190.

LINKS

Table of n, a(n) for n=0..23.

PROG

(PARI) {a(n)=local(A, m); p=5; if(n<0, 0, m=1; A=1+O(x); while(m<=n, m*=p; A=x*subst(A, x, x^p); A=(A/(1-p*A)/x)^(1/p)); polcoeff(A, n))}

CROSSREFS

Cf. A085748, A091188, A091190.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 23 2004

STATUS

approved

A091188

G.f. A(x) satisfies both A(-x)*A(x) = A(x^2) and xA(x)^2 = B(xA(x^2)) where B(x) = x*(1+x)/(1-x).

+10
3

1, 1, 1, 2, 2, 4, 5, 10, 12, 23, 31, 58, 79, 145, 207, 374, 540, 964, 1427, 2522, 3775, 6626, 10050, 17532, 26811, 46561, 71795, 124188, 192661, 332228, 518303, 891340, 1396902, 2396912, 3771822, 6459202, 10199912, 17437727, 27622807, 47152952

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

This is a special case of sequences with g.f.s that satisfy the more general functional equation xA(x)^m = B(xA(x^m)) originated by Michael Somos; some other examples are A085748, A091190 and A091200.

LINKS

Table of n, a(n) for n=0..39.

FORMULA

Given g.f. A(x), then B(x) = x * A(x^2) satisfies 0 = f(B(x), B(x^2)) were f(u, v) = u^2 * (1 - v) - v * (1 + v). - Michael Somos, Aug 02 2011

EXAMPLE

1 + x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 5*x^6 + 10*x^7 + 12*x^8 + 23*x^9 + ...

q + q^3 + q^5 + 2*q^7 + 2*q^9 + 4*q^11 + 5*q^13 + 10*q^15 + 12*q^17 + ...

PROG

(PARI) {a(n) = local(A, m); if( n<0, 0, m=1; A = 1 + O(x); while( m<=n, m*=2; A = x * subst(A, x, x^2); A = (A *(1 + A) /(1 - A) / x)^(1/2)); polcoeff(A, n))}

CROSSREFS

Cf. A085748, A091190, A091200, A092869.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 22 2004

STATUS

approved

A344613

Number of rooted binary trees (in which all inner nodes have precisely two children) with n leaves and with maximal number of cherries (i.e., maximal number of pendant subtrees with two leaves).

+10
2

1, 1, 1, 1, 2, 1, 4, 2, 9, 3, 20, 6, 46, 11, 106, 23, 248, 46, 582, 98, 1376, 207, 3264, 451, 7777, 983, 18581, 2179, 44526, 4850, 106936, 10905, 257379, 24631, 620577, 56011, 1498788, 127912, 3625026, 293547, 8779271, 676157, 21287278, 1563372, 51671864, 3626149, 125550018

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

COMMENTS

This sequence describes the number of rooted binary trees with n leaves with maximal cherry tree balance index or, equivalently, with minimal modified cherry tree balance index.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..5000

S. J. Kersting and M. Fischer, Measuring tree balance using symmetry nodes - a new balance index and its extremal properties, arXiv:2105.00719 [q-bio.PE], 2021.

FORMULA

a(n) = A001190(n/2) if n even, otherwise a(n) = A085748((n-1)/2).

MAPLE

b:= proc(n) option remember; `if`(n<2, n, `if`(n::odd, 0,

(t-> t*(1-t)/2)(b(n/2)))+add(b(i)*b(n-i), i=1..n/2))

end:

g:= proc(n) option remember; `if`(n<1, 1,

add(g(n-i)*b(i), i=1..n))

end:

a:= n-> `if`(n::even, b(n/2), g((n-1)/2)):

seq(a(n), n=1..52); # Alois P. Heinz, Jun 09 2021

MATHEMATICA

(* WE generates the Wedderburn Etherington Numbers, OEIS sequence A001190 *)

WE[n_] := Module[{i},

If[n == 1, Return[1],

If[Mod[n, 2] == 0,

Return[

WE[n/2]*(WE[n/2] + 1)/2 + Sum[WE[i]*WE[n - i], {i, 1, n/2 - 1}]],

Return[Sum[WE[i]*WE[n - i], {i, 1, Floor[n/2]}]]

]

];

(* b is just a support function *)

b[n_] := b[n] =

If[n < 2, n,

If[OddQ[n], 0, Function[t, t*(1 - t)/2][b[n/2]]] +

Sum[b[i]*b[n - i], {i, 1, n/2}]];

(* c generates the elements of OEIS sequence A085748 *)

c[n_] := c[n] = If[n < 1, 1, Sum[c[n - i]*b[i], {i, 1, n}]];

(* a generates the number of rooted binary trees with maximal number of cherries *)

a[n_] := Module[{},

If[EvenQ[n], Return[WE[n/2]], Return[c[(n - 1)/2]]]]

CROSSREFS

Cf. A001190, A085748.

KEYWORD

nonn

AUTHOR

Mareike Fischer, Jun 09 2021

STATUS

approved

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