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 1-3 of 3 results.

A019275 Incorrect version of A035010.

Original entry on oeis.org

1, 2, 4, 38, 132, 420, 1426, 4834, 16796, 58688, 208012, 742636, 2674384, 9693976, 35357670, 129641774
Offset: 1

Views

Author

Vincent Blondel (blondel(AT)ulg.ac.be)

Keywords

Comments

The term 14 is missing here.

A349450 Dirichlet inverse of right-shifted Catalan numbers [as when started from A000108(0): 1, 1, 2, 5, 14, 42, etc.].

Original entry on oeis.org

1, -1, -2, -4, -14, -38, -132, -420, -1426, -4834, -16796, -58688, -208012, -742636, -2674384, -9693976, -35357670, -129641774, -477638700, -1767253368, -6564119892, -24466233428, -91482563640, -343059494120, -1289904147128, -4861945985428, -18367353066440, -69533549429280, -263747951750360, -1002242211282032
Offset: 1

Views

Author

Antti Karttunen, Nov 22 2021

Keywords

Links

Crossrefs

Cf. A000108, A035010, A035102.
Cf. also A349449, A349451, A349452, A349453, A349564.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#] * CatalanNumber[n/# - 1] &, # < n &]; Array[a, 30] (* Amiram Eldar, Nov 22 2021 *)
  • PARI
    A000108(n) = binomial(2*n, n)/(n+1);
memoA349450 = Map();
A349450(n) = if(1==n,1,my(v); if(mapisdefined(memoA349450,n,&v), v, v = -sumdiv(n,d,if(dA000108((n/d)-1)*A349450(d),0)); mapput(memoA349450,n,v); (v)));

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A000108((n/d)-1) * a(d).
For n > 1, a(n) = -A035010(n) = A035102(n) - A000108(n-1).
G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} Catalan(k-1) * A(x^k). - Ilya Gutkovskiy, Feb 23 2022
x = Sum_{n>=1} a(n) * C(x^n) where C(x) = (1 - sqrt(1-4*x))/2 is the g.f. of the Catalan numbers (A000108). - Paul D. Hanna, Nov 27 2024

A035102 Composite binary rooted trees with external nodes.

Original entry on oeis.org

0, 0, 1, 0, 4, 0, 9, 4, 28, 0, 98, 0, 264, 56, 869, 0, 3016, 0, 9822, 528, 33592, 0, 119530, 196, 416024, 5712, 1486724, 0, 5369336, 0, 19392637, 67184, 70715340, 3696, 259535958, 0
Offset: 2

Views

Author

Bernard AMERLYNCK (B.Amerlynck(AT)ulg.ac.be)

Keywords

Comments

If a,b are binary trees, a.b is equal to tree b where a copy of a is put on each of b's external node. This is non-commutative but associative. A binary tree a is prime if it is different from the 1 node tree and if a=b.c implies that b or c is equal to the 1 node tree.

Links

Crossrefs

Cf. A035010.

Programs

  • Mathematica
    (* b = A035010 *) b[n_] := b[n] = CatalanNumber[n-1] - Sum[If[Divisible[n, d1], d2 = n/d1; b[d1]*CatalanNumber[d2-1], 0], {d1, 2, n-1}]; b[2] = 1; a[n_] := a[n] = CatalanNumber[n-1] - b[n]; Table[a[n], {n, 2, 37}] (* Jean-François Alcover, Jul 17 2012, after formula *)

Formula

A035010(n)+A035102(n)=Catalan(n-1)=A000108(n-1).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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