cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A091190 G.f. A(x) satisfies x*A(x)^3 = B(x*A(x^3)) where B(x) = x/(1 - 3*x).

Original entry on oeis.org

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
Offset: 0

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Author

Paul D. Hanna, Feb 22 2004

Keywords

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.

Examples

			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 ).

Links

Crossrefs

Cf. A085748, A091188, A091200, A370441.

Programs

  • 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 *)
  • 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),", "))

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)

Extensions

Corrected by T. D. Noe, Oct 25 2006

A091200 G.f. A(x) satisfies xA(x)^5 = B(xA(x^5)) where B(x) = x/(1-5x).

Original entry on oeis.org

1, 1, 3, 11, 44, 185, 802, 3553, 15994, 72886, 335387, 1555487, 7261310, 34083382, 160730900, 761039051, 3616102911, 17235223345, 82372594183, 394648349447, 1894921311499, 9116598414141, 43939539520427, 212124129983285
Offset: 0

Views

Author

Paul D. Hanna, Feb 23 2004

Keywords

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.

Crossrefs

Cf. A085748, A091188, A091190.

Programs

  • 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))}

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).

Original entry on oeis.org

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
Offset: 0

Views

Author

Paul D. Hanna, Feb 22 2004

Keywords

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.

Examples

			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 + ...

Crossrefs

Cf. A085748, A091190, A091200, A092869.

Programs

  • 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))}

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

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).

Original entry on oeis.org

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
Offset: 1

Views

Author

Mareike Fischer, Jun 09 2021

Keywords

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

Crossrefs

Cf. A001190, A085748.

Programs

  • 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]]]]

Formula

a(n) = A001190(n/2) if n even, otherwise a(n) = A085748((n-1)/2).
Showing 1-4 of 4 results.

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