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A110501 Unsigned Genocchi numbers (of first kind) of even index. 42
1, 1, 3, 17, 155, 2073, 38227, 929569, 28820619, 1109652905, 51943281731, 2905151042481, 191329672483963, 14655626154768697, 1291885088448017715, 129848163681107301953, 14761446733784164001387, 1884515541728818675112649, 268463531464165471482681379 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The Genocchi numbers satisfy Seidel's recurrence: for n > 1, 0 = Sum_{j=0..floor(n/2)} (-1)^j*binomial(n, 2*j)*a(n-j). - Ralf Stephan, Apr 17 2004

The (n+1)st Genocchi number is the number of Dumont permutations of the first kind on 2n letters. In a Dumont permutation of first kind, each even integer must be followed by a smaller integer and each odd integer is either followed by a larger integer or is the last element. - Ralf Stephan, Apr 26 2004

The (n+1)-st Genocchi number is also the number of ways to place n rooks (attacking along planes; also called super rooks of power 2 by Golomb and Posner) on the three-dimensional Genocchi boards of size n. The Genocchi board of size n consists of cells of the form (i, j, k) where min{i, j} <= k and 1 <= k <= n. A rook placement on this board can also be realized as a pair of permutations of n the smallest number in the i-th position of the two permutations is not larger than i. - Feryal Alayont, Nov 03 2012

REFERENCES

L. Carlitz, A conjecture concerning Genocchi numbers. Norske Vid. Selsk. Skr. (Trondheim) 1971, no. 9, 4 pp. MR0297697 (45 #6749)

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.

Leonhard Euler, Institutionum Calculi Differentialis, volume 2 (1755), para. 181.

A. Genocchi, Intorno all'espressione generale de'numeri Bernulliani, Ann. Sci. Mat. Fis., 3 (1852), 395-405.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2 (1999) p. 74; see Problem 5.8.

H. M. Terrill and E. M. Terrill, Tables of numbers related to the tangent coefficients, J. Franklin Inst., 239 (1945), 64-67.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..100

F. Alayont and N. Krzywonos, Rook Polynomials in Three and Higher Dimensions, to appear in Involve

F. Alayont, R. Moger-Reischer and R. Swift, Rook Number Interpretations of Generalized Central Factorial and Genocchi Numbers, preprint, 2012.

R. C. Archibald, Review of Terrill-Terrill paper, Math. Comp., 1 (1945), pp. 385-386.

P. Bala, A triangle for calculating the Genocchi numbers

Ange Bigeni, A bijection between the irreducible k-shapes and the surjective pistols of height k-1, arXiv preprint arXiv:1402.1383 [math.CO] (2014). Also Discrete Math., 338 (2015), 1432-1448.

Ange Bigeni, Enumerating the symplectic Dellac configurations, arXiv:1705.03804 [math.CO], 2017.

Ange Bigeni, The universal sl2 weight system and the Kreweras triangle, arXiv:1712.05475 [math.CO], 2017.

Ange Bigeni, Combinatorial interpretations of the Kreweras triangle in terms of subset tuples, arXiv:1712.01929 [math.CO], 2017.

E. Clark and R. Ehrenborg, The excedance algebra, Discr. Math., 313 (2013), 1429-1435.

D. Dumont, Sur une conjecture de Gandhi concernant les nombres de Genocchi, Discrete Mathematics 1 (1972) 321-327.

D. Dumont, Interpretations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.

Dominique Dumont & Dominique Foata, Une propriété de symétrie des nombres de Genocchi Bull. Soc. Math. France 104 (1976), no. 4, 433-451. MR0434830 (55 #7794)

D. Dumont & G. Viennot, A combinatorial interpretation of the Seidel generation of Genocchi numbers, Preprint, Annotated scanned copy.

Dominique Dumont & Gérard Viennot, A combinatorial interpretation of the Seidel generation of Genocchi numbers, Combinatorial mathematics, optimal designs and their applications (Proc. Sympos. Combin. Math. and Optimal Design, Colorado State Univ., Fort Collins, Colo., 1978). Ann. Discrete Math. 6 (1980), 77-87. MR0593524 (82j:10024).

A. L. Edmonds and S, Klee, The combinatorics of hyperbolized manifolds, arXiv preprint arXiv:1210.7396 [math.CO], 2012.

Richard Ehrenborg & Einar Steingrímsson, Yet another triangle for the Genocchi numbers, European J. Combin. 21 (2000), no. 5, 593-600. MR1771988 (2001h:05008)

J. M. Gandhi, Research Problems: A Conjectured Representation of Genocchi Numbers, Amer. Math. Monthly 77 (1970), no. 5, 505-506. MR1535914

S. W. Golomb and E. C. Posner, Rook Domains, Latin Squares, Affine Planes, and Error-Distributing Codes, Transactions of the Information Theory Group of the IEEE, Vol. 10, No. 3 (1964), 196-208.

Guo-Niu Han, Enumeration of Standard Puzzles

Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]

Guo-Niu Han, Jing-Yi Liu, Divisibility properties of the tangent numbers and its generalizations, arXiv:1707.08882 [math.CO], 2017.

Florent Hivert and Olivier Mallet, Combinatorics of k-shapes and Genocchi numbers, in FPSAC 2011, Reykjav´k, Iceland DMTCS proc. AO, 2011, 493-504.

John Riordan & Paul R. Stein, Proof of a conjecture on Genocchi numbers, Discrete Math. 5 (1973), 381-388. MR0316372 (47 #4919) - From N. J. A. Sloane, Jun 12 2012

FORMULA

(-1)^n * a(n) = A036968(2*n) = A001469(n).

a(n) = 2*(-1)^n*(1-4^n)*B_{2*n} (B = A027641/A027642 are Bernoulli numbers).

A002105(n) = 2^(n-1)/n * a(n). - Don Knuth, Jan 16 2007

E.g.f.: x * tan(x/2) = Sum_{k > 0} a(k) * x^(2*k) / (2*k)!.

E.g.f.: x * tan(x/2) = x^2 / (2 - x^2 / (6 - x^2 / (... 4*k+2 - x^2 / (...)))). - Michael Somos, Mar 13 2014

O.g.f.: Sum_{n >= 0} n!^2 * x^(n+1) / Product_{k = 1..n} (1 + k^2*x). - Paul D. Hanna, Jul 21 2011

a(n) = Sum_{k = 0..2*n} (-1)^(n-k+1)*Stirling2(2*n, k)*A059371(k). - Vladeta Jovovic, Feb 07 2004

O.g.f.: A(x) = x/(1-x/(1-2*x/(1-4*x/(1-6*x/(1-9*x/(1-12*x/(... -[(n+1)/2]*[(n+2)/2]*x/(1- ...)))))))) (continued fraction). - Paul D. Hanna, Jan 16 2006

a(n) = Pi^(-2*n)*integral(log(t/(1-t))^(2*n)-log(1-1/t)^(2*n) dt,t=0,1). - Gerry Martens, May 25 2011

a(n) = the upper left term of M^(n-1); M is an infinite square production matrix with M[i,j] = C(i+1,j-1), i.e. Pascal's triangle without the first two rows and right border, see the examples and Maple program. - Gary W. Adamson, Jul 19 2011

G.f.: 1/U(0) where U(k) = 1 + 2*(k^2)*x - x*((k+1)^2)*(x*(k^2)+1)/U(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Sep 15 2012

a(n+1) = Sum_{k=0..n} A211183(n, k)*2^(n-k). - Philippe Deléham, Feb 03 2013

G.f.: 1 + x/(G(0)-x) where G(k) =  2*x*(k+1)^2 + 1 - x*(k+2)^2*(x*k^2+2*x*k+x+1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Feb 10 2013

G.f.: G(0) where G(k) = 1 + x*(2*k+1)^2/( 1 + x + 4*x*k + 4*x*k^2 - 4*x*(k+1)^2*(1 + x + 4*x*k + 4*x*k^2)/(4*x*(k+1)^2 + (1 + 4*x + 8*x*k + 4*x*k^2)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Feb 11 2013

G.f.: R(0), where R(k) = 1 - x*(k+1)^2/( x*(k+1)^2 - 1/(1 - x*(k+1)*(k+2)/( x*(k+1)*(k+2) - 1/R(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Oct 27 2013

E.g.f. (offset 1): sqrt(x)*tan(sqrt(x)/2) = Q(0)*x/2, where Q(k) = 1 - x/(x - 4*(2*k+1)*(2*k+3)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 06 2014

Pi^2/6 = 2*Sum_{k=1..N} (-1)^(k-1)/k^2 + (-1)^N/N^2(1 - 1/N + 1/N^3 - 3/N^5 + 17/N^7 - 155/N^9 +- ...), where the terms in the parenthesis are (-1)^n*a(n)/N^(2n-1). - M. F. Hasler, Mar 11 2015

a(n) = 2*n*|euler(2*n-1, 0)|. - Peter Luschny, Jun 09 2016

a(n) = 4^(1-n) * (4^n-1) * Pi^(-2*n) * (2*n)! * zeta(2*n). - Daniel Suteu, Oct 14 2016

EXAMPLE

E.g.f.: x*tan(x/2) = x^2/2! + x^4/4! + 3*x^6/6! + 17*x^8/8! + 155*x^10/10! + ...

O.g.f.: A(x) = x + x^2 + 3*x^3 + 17*x^4 + 155*x^5 + 2073*x^6 + ...

where A(x) = x + x^2/(1+x) + 2!^2*x^3/((1+x)*(1+4*x)) + 3!^2*x^4/((1+x)*(1+4*x)*(1+9*x)) + 4!^2*x^5/((1+x)*(1+4*x)*(1+9*x)*(1+16*x)) + ... . - Paul D. Hanna, Jul 21 2011

From Gary W. Adamson, Jul 19 2011: (Start)

The first few rows of production matrix M are:

1, 2,  0,  0,  0, 0, ...

1, 3,  3,  0,  0, 0, ...

1, 4,  6,  4,  0, 0, ...

1, 5, 10, 10,  5, 0, ...

1, 6, 15, 20, 15, 6, ... (End)

MAPLE

A110501 := proc(n)

    2*(-1)^n*(1-4^n)*bernoulli(2*n) ;

end proc:

seq(A110501(n), n=0..10) ; # R. J. Mathar, Aug 02 2013

MATHEMATICA

a[n_] := 2*(4^n - 1) * BernoulliB[2n] // Abs; Table[a[n], {n, 19}] (* Jean-François Alcover, May 23 2013 *)

PROG

(PARI) {a(n) = if( n<1, 0, 2 * (-1)^n * (1 - 4^n) * bernfrac( 2*n))};

(PARI) {a(n) = if( n<1, 0, (2*n)! * polcoeff( x * tan(x/2 + x * O(x^(2*n))), 2*n))};

(PARI) {a(n)=polcoeff(sum(m=0, n, m!^2*x^(m+1)/prod(k=1, m, 1+k^2*x+x*O(x^n))), n)} /* Paul D. Hanna, Jul 21 2011 */

(Sage) # Algorithm of L. Seidel (1877)

# n -> [a(1), ..., a(n)] for n >= 1.

def A110501_list(n) :

    D = []; [D.append(0) for i in (0..n+2)]; D[1] = 1

    R = [] ; b = True

    for i in(0..2*n-1) :

        h = i//2 + 1

        if b :

            for k in range(h-1, 0, -1) : D[k] += D[k+1]

        else :

            for k in range(1, h+1, 1) :  D[k] += D[k-1]

        b = not b

        if b : R.append(D[h])

    return R

A110501_list(19) # Peter Luschny, Apr 01 2012

(MAGMA) [Abs(2*(4^n-1)*Bernoulli(2*n)): n in [1..20]]; // Vincenzo Librandi, Jul 28 2017

CROSSREFS

Cf. A001469, A002105, A036968, A211183.

Sequence in context: A135751 A168441 A001469 * A274539 A066211 A163884

Adjacent sequences:  A110498 A110499 A110500 * A110502 A110503 A110504

KEYWORD

nonn

AUTHOR

Michael Somos, Jul 23 2005

EXTENSIONS

Edited by M. F. Hasler, Mar 22 2015

STATUS

approved

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Last modified May 25 08:24 EDT 2018. Contains 304551 sequences. (Running on oeis4.)