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A005437 Column of Kempner tableau.

%I A005437 M1276 #43 Apr 28 2023 12:09:44
%S A005437 1,1,1,2,4,14,46,224,1024,6320,36976,275792,1965664,17180144,
%T A005437 144361456,1446351104,13997185024,158116017920,1731678144256,
%U A005437 21771730437632,266182076161024,3686171162253824,49763143319190016,752594181757712384,11118629668610842624
%N A005437 Column of Kempner tableau.
%C A005437 From _Peter Luschny_, Jul 09 2012: (Start)
%C A005437 Also the central column of the Seidel-Entringer triangles A008281 and A008282.
%C A005437 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 A005437 In Gelineau, Shin, and Zeng (section 6.1) twelve interpretations of the numbers can be found. (End)
%C A005437 This sequence is the central sequence of numbers in the following table:
%C A005437 A_0               1
%C A005437 B_1               1   0
%C A005437 A_2           0   1   1
%C A005437 B_3           2   2   1   0
%C A005437 A_4       0   2   4   5   5
%C A005437 B_5      16  16  14  10   5  0
%C A005437 A_6   0  16  32  46  56  61 61
%C A005437 B_7 272 272 256 224 178 122 61 0
%C A005437 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 A005437 Named after the English-American mathematician Aubrey John Kempner (1880-1973). - _Amiram Eldar_, Jun 23 2021
%D A005437 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005437 Alois P. Heinz, <a href="/A005437/b005437.txt">Table of n, a(n) for n = 0..485</a>
%H A005437 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 A005437 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 A005437 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 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 A005437 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 A005437 Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Nov 12 2018, after _Peter Luschny_ *)
%Y A005437 Cf. A008281, A008282, A010094, A108040.
%Y A005437 Main diagonal of A064192.
%K A005437 nonn
%O A005437 0,4
%A A005437 _Simon Plouffe_
%E A005437 More terms from _Sean A. Irvine_, Jun 25 2016
%E A005437 Offset set to 0 by _Peter Luschny_, Oct 15 2018