A162470 Table read by rows: the Catalan number C(n) followed by pairs (C(n-1-j), C(j+1)), j=0 to n-2.

1, 2, 1, 1, 5, 2, 1, 1, 2, 14, 5, 1, 2, 2, 1, 5, 42, 14, 1, 5, 2, 2, 5, 1, 14, 132, 42, 1, 14, 2, 5, 5, 2, 14, 1, 42, 429, 132, 1, 42, 2, 14, 5, 5, 14, 2, 42, 1, 132, 1430, 429, 1, 132, 2, 42, 5, 14, 14, 5, 42, 2, 132, 1, 429, 4862, 1430, 1, 429, 2, 132, 5, 42, 14, 14, 42, 5, 132, 2, 429, 1, 1430

Offset

1,2

Comments

The table starts row n with A000108(n) = C(n) and then visualizes the convolution formula C(n) = Sum_{j=0..n-1} C(j)*C(n-1-j) by listing some pairs of that structure in the same row.

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Example

Starting in row n=1 the table reads
  1;
  2,1,1;
  5,2,1,1,2;
  14,5,1,2,2,1,5;
  42,14,1,5,2,2,5,1,14;
  132,42,1,14,2,5,5,2,14,1,42;
The row n=3 represents 5=2*1+1*1+1*2, for example, where the middle term is not represented in the table.

Mathematica

A162470row[n_] := Riffle[CatalanNumber[Range[n, 1, -1]], CatalanNumber[Range[n - 1]], {3, -1, 2}];
Array[A162470row, 10] (* Paolo Xausa, Dec 29 2025 *)

Crossrefs

Cf. A000108.

Sequence in context: A022661 A120292 A179318 * A343234 A174004 A128604

Adjacent sequences: A162467 A162468 A162469 * A162471 A162472 A162473

Keyword

nonn,tabf,easy,less

Author

Juri-Stepan Gerasimov, Jul 04 2009

Extensions

Definition and comment rephrased by R. J. Mathar, Sep 16 2009

a(10) = 5 removed and more terms from Paolo Xausa, Dec 29 2025

Status

approved