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A000366
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Genocchi numbers of second kind (A005439) divided by 2^(n-1).
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6
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1, 1, 2, 7, 38, 295, 3098, 42271, 726734, 15366679, 391888514, 11860602415, 420258768950, 17233254330343, 809698074358250, 43212125903877439, 2599512037272630686, 175079893678534943287, 13122303354155987156306
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| The earliest known reference to these numbers is the Dellac Marseille memoir. - D. E. Knuth, Jul 11 2007
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.
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.
The entry for A005439 has further information.
The number of sequences (I_1,...,I_{n-1}) consisting of subsets of the set {1,...,n} such that the number of elements in I_k is exactly k and I_k\subset I_{k+1}\cup {k+1}. The Euler characteristics of the degenerate flag varieties of type A. - Evgeny Feigin, Dec 15 2011
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REFERENCES
| Anonymous, l'Intermediaire des Math\'ematiciens, 7 (1900), p. 328.
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.
Hippolyte Dellac, Note sur l'\'elimination, m\'ethode de parall\'elogramme, Annales de la Facult\'e des Sciences de Marseille, XI (1901), 141-164.
Hippolyte Dellac, Problem 1735, L'Interm\'{e}diaire des Math\'{e}maticiens, Vol. 7 (1900), 9-10.
E. Feigin, The median Genocchi numbers, Q-analogues and continued fractions, arXiv:1111.0740.
E. Feigin, Degenerate flag varieties and the median Genocchi numbers, arXiv:1101.1898,
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
G. Kreweras, Sur les permutations compte'es par les nombres de Genocchi..., Europ. J. Comb., vol. 18, pp. 49-58, 1997.
Kreweras, G.; and Dumont, D.; Sur les anagrammes alternes. Discrete Math. 211, No.1-3, 103-110 (2000). Zbl 0944.05003
E. Lemoine, l'Intermediaire des Math\'ematiciens, 8 (1901), 168-169.
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.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..100
I. M. Gessel, Applications of the classical umbral calculus.
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FORMULA
| 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:
For example, the first four terms on the right are
1
... 2x - 2x^2 + 2x^3 + ...
........ 9x^2 - 36x^3 + ...
............... 72x^3 + ...
summing to 1+2x+7x^2+38x^3+... . Of course one can replace x by 2x and get a generating function for A005439. (End)
(-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.
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
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))).
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MATHEMATICA
| a[n_] = (-2^(-1))^(n-2)* Sum[ Binomial[n, k]*(1 - 2^(n+k+1))*BernoulliB[n+k+1], {k, 0, n}]; Table[a[n], {n, 19}] (* From Jean-François Alcover, Jul 18 2011, after PARI prog. *)
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PROG
| (PARI) a(n)=(-1/2)^(n-2)*sum(k=0, n, binomial(n, k)*(1-2^(n+k+1))*bernfrac(n+k+1))
(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)
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CROSSREFS
| Cf. A001469, A005439, A130168, A130169.
First column, first diagonal and row sums of triangle A014784.
Sequence in context: A084552 A094664 A001858 * A106211 A014058 A119602
Adjacent sequences: A000363 A000364 A000365 * A000367 A000368 A000369
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KEYWORD
| nonn,easy,nice
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AUTHOR
| D. E. Knuth, N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from David W. Wilson (davidwwilson(AT)comcast.net), Jan 11 2001
Edited by Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 17 2004
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