A005437 Column of Kempner tableau.

1, 1, 1, 2, 4, 14, 46, 224, 1024, 6320, 36976, 275792, 1965664, 17180144, 144361456, 1446351104, 13997185024, 158116017920, 1731678144256, 21771730437632, 266182076161024, 3686171162253824, 49763143319190016, 752594181757712384, 11118629668610842624

Offset: 0

Author

Simon Plouffe

Keywords

• nonn

Comments

From Peter Luschny, Jul 09 2012: (Start)

Also the central column of the Seidel-Entringer triangles A008281 and A008282.

a(n) takes alternatingly the values of the central column of the Seidel-Entriger triangles A008281 (1,1,4,46,...) and A008282 (1,2,14,224,..).

In Gelineau, Shin, and Zeng (section 6.1) twelve interpretations of the numbers can be found. (End)

This sequence is the central sequence of numbers in the following table:

A_0 1

B_1 1 0

A_2 0 1 1

B_3 2 2 1 0

A_4 0 2 4 5 5

B_5 16 16 14 10 5 0

A_6 0 16 32 46 56 61 61

B_7 272 272 256 224 178 122 61 0

where row A_k is obtained from row B_(k-1) by the sequence 0, b_1, b_1+b_2, ..., b_1+b_2+....+b_k and row B_k is obtained from the row A_(k-1) by the sequence a_1+a_2+....+a_k, ..., a_(k-1)+a_k, a_k, 0. - Sean A. Irvine, Jun 25 2016

Named after the English-American mathematician Aubrey John Kempner (1880-1973). - Amiram Eldar, Jun 23 2021

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..485
• Yoann Gelineau, Heesung Shin and Jiang Zeng, Bijections for Entringer families, hal-00472187, 2010.
• Yoann Gelineau, Heesung Shin and Jiang Zeng, Bijections for Entringer families, arXiv:1004.2179 [math.CO], 2010.
• Gérard Viennot, Interprétations combinatoires des nombres d'Euler et de Genocchi, Séminaire de théorie des nombres, 1980/1981, Exp. No. 11, p. 41, Univ. Bordeaux I, Talence, 1982.

Crossrefs

Cf. A008281, A008282, A010094, A108040.

Main diagonal of A064192.

Programs

Maple

A005437 := proc(n) local S; S := proc(n, k) option remember; if k=0 then `if`(n=0, 1, 0) else S(n, k-1)+S(n-1, n-k) fi end: S(n, iquo(n+1, 2)) end; seq(A005437(i), i=0..24); # Peter Luschny, Jul 09 2012

Mathematica

a[n_] := Module[{S}, S[m_, k_] := S[m, k] = If[k == 0, If[m == 0, 1, 0], S[m, k-1] + S[m-1, m-k]]; S[n, Quotient[n+1, 2]]];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Nov 12 2018, after Peter Luschny *)

Extensions

More terms from Sean A. Irvine, Jun 25 2016

Offset set to 0 by Peter Luschny, Oct 15 2018