1,3

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.

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.

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.

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.

T. D. Noe, Table of n, a(n) for n=1..100

I. M. Gessel, Applications of the classical umbral calculus.

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))).

(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)

Cf. A001469, A005439, A130168, A130169.

First column, first diagonal and row sums of triangle A014784.

Adjacent sequences: A000363 A000364 A000365 this_sequence A000367 A000368 A000369

Sequence in context: A084552 A094664 A001858 this_sequence A106211 A014058 A119602

nonn,easy,nice

D. E. Knuth, njas

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