Search: id:A000366
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%I A000366
%S A000366 1,1,2,7,38,295,3098,42271,726734,15366679,391888514,11860602415,
%T A000366 420258768950,17233254330343,809698074358250,43212125903877439,
%U A000366 2599512037272630686,175079893678534943287,13122303354155987156306
%N A000366 Genocchi numbers of second kind (A005439) divided by 2^(n-1).
%C A000366 The earliest known reference to these numbers is the Dellac Marseille 
               memoir. - D. E. Knuth, Jul 11 2007
%C A000366 According to Ira Gessel, Dellac's interpretation is the following: start 
               with a 2n X n array of cells and consider the set D of cells in rows 
               i through i+n of column i, for i from 1 to n. Then a(n) is the number 
               of subsets of D containing two cells in each column and one cell 
               in each row.
%C A000366 Barsky proved that for even n>1, a(n) is congruent to 3 mod 4 and for 
               odd n>1, congruent to 2 mod 4. Gessel shows that for even n>5, a(n) 
               is congruent to 4n-1 mod 16 and for odd n>2 that a(n)/2 is congruent 
               to 2-n mod 8.
%C A000366 The entry for A005439 has further information.
%D A000366 Anonymous, l'Intermediaire des Math\'ematiciens, 7 (1900), p. 328.
%D A000366 D. Barsky, Congruences pour les nombres de Genocchi de 2e espece, Groupe 
               d'etude d'Analyse ultrametrique, 8e annee, no. 34, 1980/81, 13 pp.
%D A000366 Hippolyte Dellac, Note sur l'\'elimination, m\'ethode de parall\'elogramme, 
               Annales de la Facult\'e des Sciences de Marseille, XI (1901), 141-164.
%D A000366 Hippolyte Dellac, Problem 1735, L'Interm\'{e}diaire des Math\'{e}maticiens, 
               Vol. 7 (1900), 9-10.
%D A000366 G. Han and J. Zeng, On a q-sequence that generalizes the median Genocchi 
               numbers, Annal Sci. Math. Quebec, 23(1999), no. 1, 63-72
%D A000366 G. Kreweras, Sur les permutations compte'es par les nombres de Genocchi..., 
               Europ. J. Comb., vol. 18, pp. 49-58, 1997.
%D A000366 E. Lemoine, l'Intermediaire des Math\'ematiciens, 8 (1901), 168-169.
%D A000366 L. Seidel, Ueber eine einfache Entstehungsweise der Bernoulli'schen Zahlen 
               und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen 
               Classe der k\"oniglich bayerischen Akademie der Wissenschaften zu 
               M\"unchen, volume 7 (1877), 157-187.
%H A000366 T. D. Noe, <a href="b000366.txt">Table of n, a(n) for n=1..100</a>
%H A000366 I. M. Gessel, <a href="http://www.arXiv.org/abs/math.CO/0108121">Applications 
               of the classical umbral calculus</a>.
%F A000366 Comment from D. E. Knuth, Jul 11 2007: (Start) The anonymous 1900 note 
               in Interm. Math. gives a formula that is equivalent to a nice generating 
               function:
%F A000366 For example, the first four terms on the right are
%F A000366 1
%F A000366 ... 2x - 2x^2 + 2x^3 + ...
%F A000366 ........ 9x^2 - 36x^3 + ...
%F A000366 ............... 72x^3 + ...
%F A000366 summing to 1+2x+7x^2+38x^3+... . Of course one can replace x by 2x and 
               get a generating function for A005439. (End)
%F A000366 (-2)^(2-n) * sum{k=0..n, C(n, k)*(1-2^(n+k+1))*B(n+k+1)}, with B(n) the 
               Bernoulli numbers.
%F A000366 O.g.f.: A(x) = x/(1-x/(1-x/(1-3*x/(1-3*x/(1-6*x/(1-6*x/(... -[n/2+1]*[n/
               2+2]/2*x/(1- ...)))))))) (continued fraction). - Paul D. Hanna (pauldhanna(AT)juno.com), 
               Oct 07 2005
%F A000366 Sum_{n>0} a(n)x^n = Sum_{n>0} (n!^2/2^{n-1}) (x^n/((1+x)(1+3x)...(1+binomial(n,
               2)x))).
%o A000366 (PARI) a(n)=(-1/2)^(n-2)*sum(k=0,n,binomial(n,k)*(1-2^(n+k+1))*bernfrac(n+k+1))
%o A000366 (PARI) {a(n)=local(CF=1+x*O(x^n));if(n<1,return(0), for(k=1,n,CF=1/(1-((n-k)\2+1)*((n-k)\2+2)/
               2*x*CF));return(Vec(CF)[n]))} (Hanna)
%Y A000366 Cf. A001469, A005439, A130168, A130169.
%Y A000366 First column, first diagonal and row sums of triangle A014784.
%Y A000366 Sequence in context: A084552 A094664 A001858 this_sequence A106211 A014058 
               A119602
%Y A000366 Adjacent sequences: A000363 A000364 A000365 this_sequence A000367 A000368 
               A000369
%K A000366 nonn,easy,nice
%O A000366 1,3
%A A000366 D. E. Knuth, N. J. A. Sloane (njas(AT)research.att.com).
%E A000366 More terms from David W. Wilson (davidwwilson(AT)comcast.net), Jan 11 
               2001
%E A000366 Edited by Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 17 2004

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