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A278763 Triangular array: row n shows the number of edges in successive levels of a graph of the partitions of n; see Comments. 2
1, 1, 1, 1, 2, 2, 1, 2, 4, 2, 1, 2, 5, 6, 3, 1, 2, 5, 8, 9, 3, 1, 2, 5, 9, 14, 12, 4, 1, 2, 5, 9, 16, 20, 16, 4, 1, 2, 5, 9, 17, 25, 30, 20, 5, 1, 2, 5, 9, 17, 27, 39, 40, 25, 5, 1, 2, 5, 9, 17, 28, 44, 56, 55, 30, 6, 1, 2, 5, 9, 17, 28, 46, 65, 80, 70, 36 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

The k-th number in row n (with rows numbered 2,3,4,...) is the number of edges from partitions of n into k parts to partitions of n into k-1 parts, for k = n..2, where partitions p and q share an edge if p has one more part than q, and exactly one part of p is a sum of two parts of q. The limiting row is A000097, which also gives the row sums.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..1000

EXAMPLE

First 9 rows (for n = 2 to 10):

  1;

  1,  1;

  1,  2,  2;

  1,  2,  4,  2;

  1,  2,  5,  6,  3;

  1,  2,  5,  8,  9,  3;

  1,  2,  5,  9, 14, 12,  4;

  1,  2,  5,  9, 16, 20, 16,  4;

  1,  2,  5,  9, 17, 25, 30, 20,  5;

  1,  2,  5,  9, 17, 27, 39, 40, 25,  5;

(See also the Example at A278762, for n = 5.)

MATHEMATICA

p[n_] := p[n] = IntegerPartitions[n];

s[n_, k_] := s[n, k] = Select[p[n], Length[#] == k &];

x[n_, k_] := x[n, k] = Map[Length, Map[Union, s[n, k]]];

b[h_] := b[h] = h (h - 1)/2;

e[n_, k_] := e[n, k] = Total[Map[b, x[n, k]]];

Flatten[Table[e[n, k], {n, 2, 20}, {k, 2, n - 1}]]  (* A278762 sequence *)

TableForm[Table[e[n, k], {n, 2, 20}, {k, 2, n - 1}]]  (* A278762 triangle *)

Flatten[Table[e[n, k], {n, 2, 20}, {k, n - 1, 2, -1}]]  (* A278763 sequence *)

TableForm[Table[e[n, k], {n, 2, 20}, {k, n - 1, 2, -1}]]  (* A278763 triangle *)

CROSSREFS

Cf. A000041, A000097 (row sums), A278762.

Sequence in context: A228441 A156260 A056671 * A278762 A055076 A069780

Adjacent sequences:  A278760 A278761 A278762 * A278764 A278765 A278766

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Nov 30 2016

STATUS

approved

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Last modified October 17 18:19 EDT 2018. Contains 316292 sequences. (Running on oeis4.)