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.

A300652 Number of enriched p-trees of weight 2n + 1 in which all outdegrees and all leaves are odd.

Original entry on oeis.org

1, 2, 4, 12, 40, 136, 496, 1952, 7488, 30368, 123456, 512384, 2129664, 9068672, 38391552, 165642752, 713405952, 3109135872, 13528865792, 59591322624, 261549260800, 1159547047936, 5131968999424, 22883893137408, 101851069587456, 456703499042816, 2042949493276672
Offset: 0

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Author

Gus Wiseman, Mar 10 2018

Keywords

Comments

An enriched p-tree of weight n > 0 is either a single node of weight n, or a finite sequence of at least two enriched p-trees whose weights are weakly decreasing and sum to n.

Examples

			The a(3) = 12 trees:
7,
(511), (331),
((111)31), (3(111)1), ((311)11), (31111),
((111)(111)1), (((111)11)11), ((11111)11), ((111)1111), (1111111).
		

Crossrefs

Programs

  • Mathematica
    r[n_]:=r[n]=If[OddQ[n],1,0]+Sum[Times@@r/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&OddQ[Length[#]]&]}];
    Table[r[n],{n,1,40,2}]
  • PARI
    seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^(2*k-1) + O(x^(2*n))) - 1/prod(k=1, n-1, 1 + v[k]*x^(2*k-1) + O(x^(2*n))), 2*n-1)/2); v} \\ Andrew Howroyd, Aug 26 2018

Formula

a(n) = (1 - (-1)^n)/2 + Sum_y Product_{i in y} a(i) where the sum is over all non-singleton integer partitions of n with an odd number of parts.