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Column of Kempner tableau.
(Formerly M1276)
5

%I M1276 #43 Apr 28 2023 12:09:44

%S 1,1,1,2,4,14,46,224,1024,6320,36976,275792,1965664,17180144,

%T 144361456,1446351104,13997185024,158116017920,1731678144256,

%U 21771730437632,266182076161024,3686171162253824,49763143319190016,752594181757712384,11118629668610842624

%N Column of Kempner tableau.

%C From _Peter Luschny_, Jul 09 2012: (Start)

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

%C 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,..).

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

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

%C A_0 1

%C B_1 1 0

%C A_2 0 1 1

%C B_3 2 2 1 0

%C A_4 0 2 4 5 5

%C B_5 16 16 14 10 5 0

%C A_6 0 16 32 46 56 61 61

%C B_7 272 272 256 224 178 122 61 0

%C 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

%C Named after the English-American mathematician Aubrey John Kempner (1880-1973). - _Amiram Eldar_, Jun 23 2021

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

%H Alois P. Heinz, <a href="/A005437/b005437.txt">Table of n, a(n) for n = 0..485</a>

%H Yoann Gelineau, Heesung Shin and Jiang Zeng, <a href="http://hal.archives-ouvertes.fr/hal-00472187">Bijections for Entringer families</a>, hal-00472187, 2010.

%H Yoann Gelineau, Heesung Shin and Jiang Zeng, <a href="https://arxiv.org/abs/1004.2179">Bijections for Entringer families</a>, arXiv:1004.2179 [math.CO], 2010.

%H Gérard Viennot, <a href="http://www.jstor.org/stable/44165433">Interprétations combinatoires des nombres d'Euler et de Genocchi</a>, Séminaire de théorie des nombres, 1980/1981, Exp. No. 11, p. 41, Univ. Bordeaux I, Talence, 1982.

%p 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

%t 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]]];

%t Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Nov 12 2018, after _Peter Luschny_ *)

%Y Cf. A008281, A008282, A010094, A108040.

%Y Main diagonal of A064192.

%K nonn

%O 0,4

%A _Simon Plouffe_

%E More terms from _Sean A. Irvine_, Jun 25 2016

%E Offset set to 0 by _Peter Luschny_, Oct 15 2018