This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A110501 #307 Aug 30 2025 04:41:56
%S A110501 1,1,3,17,155,2073,38227,929569,28820619,1109652905,51943281731,
%T A110501 2905151042481,191329672483963,14655626154768697,1291885088448017715,
%U A110501 129848163681107301953,14761446733784164001387,1884515541728818675112649,268463531464165471482681379
%N A110501 Unsigned Genocchi numbers (of first kind) of even index.
%C A110501 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
%C A110501 The (n+1)-st Genocchi number is the number of Dumont permutations of the first kind on 2n letters. In a Dumont permutation of the 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
%C A110501 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
%C A110501 The (n+1)-st Genocchi number is also the number of Dumont permutations of the second kind, third kind, and fourth kind on 2n letters. In a Dumont permutation of the second kind, all odd positions are weak excedances and all even positions are deficiencies. In a Dumont permutation of the third kind, all descents are from an even value to an even value. In a Dumont permutation of the fourth kind, all deficiencies are even values at even positions. - _Alexander Burstein_, Jun 21 2019
%C A110501 The (n+1)-st Genocchi number is also the number of semistandard Young tableaux of skew shape (n+1,n,...,1)/(n-1,n-2,...,1) such that the entries in row i are at most i for i=1,...,n+1. - _Alejandro H. Morales_, Jul 26 2020
%C A110501 The (n+1)-st Genocchi number is also the number of positive terms of the Okounkov-Olshanski formula for the number of standard tableaux of skew shape (n+1,n,n-1,...,1)/(n-1,n-2,...,1), given by the (2n+1)-st Euler number A000111. - _Alejandro H. Morales_, Jul 26 2020
%C A110501 The (n+1)-st Genocchi number is also the number of collapsed permutations in (2n-1) letters. A permutation pi of size 2n-1 is said to be collapsed if ceil(k/2) <= pi^{-1}(k) <= n + floor(k/2). There are 3 collapsed permutations of size 3, namely 123, 132 and 213. - _Arvind Ayyer_, Oct 23 2020
%D A110501 L. Carlitz, A conjecture concerning Genocchi numbers. Norske Vid. Selsk. Skr. (Trondheim) 1971, no. 9, 4 pp. MR0297697 (45 #6749)
%D A110501 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
%D A110501 Leonhard Euler, Institutionum Calculi Differentialis, volume 2 (1755), para. 181.
%D A110501 A. Genocchi, Intorno all'espressione generale de'numeri Bernulliani, Ann. Sci. Mat. Fis., 3 (1852), 395-405.
%D A110501 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2 (1999) p. 74; see Problem 5.8.
%H A110501 Alan Sokal, <a href="/A110501/b110501.txt">Table of n, a(n) for n = 1..250</a> (terms up to a(100) from Alois P. Heinz)
%H A110501 F. Alayont and N. Krzywonos, <a href="http://faculty.gvsu.edu/alayontf/notes/rook_polynomials_higher_dimensions_preprint.pdf">Rook Polynomials in Three and Higher Dimensions</a>, to appear in Involve
%H A110501 F. Alayont, R. Moger-Reischer and R. Swift, <a href="http://faculty.gvsu.edu/alayontf/notes/generalized_central_genocchi_numbers_preprint.pdf">Rook Number Interpretations of Generalized Central Factorial and Genocchi Numbers</a>, preprint, 2012.
%H A110501 R. C. Archibald, <a href="http://dx.doi.org/10.1090/S0025-5718-45-99088-0">Review of Terrill-Terrill paper</a>, Math. Comp., 1 (1945), pp. 385-386.
%H A110501 A. Ayyer, D. Hathcock and P. Tetali, <a href="https://arxiv.org/abs/2010.11236">Toppleable Permutations, Excedances and Acyclic Orientations</a>, arXiv:2010.11236 [math.CO], 2020.
%H A110501 Peter Bala, <a href="/A110501/a110501_1.pdf">A triangle for calculating the Genocchi numbers</a>
%H A110501 Ange Bigeni, <a href="http://arxiv.org/abs/1402.1383">A bijection between the irreducible k-shapes and the surjective pistols of height k-1</a>, arXiv preprint arXiv:1402.1383 [math.CO] (2014). Also Discrete Math., 338 (2015), 1432-1448.
%H A110501 Ange Bigeni, <a href="https://arxiv.org/abs/1705.03804">Enumerating the symplectic Dellac configurations</a>, arXiv:1705.03804 [math.CO], 2017.
%H A110501 Ange Bigeni, <a href="https://arxiv.org/abs/1712.05475">The universal sl2 weight system and the Kreweras triangle</a>, arXiv:1712.05475 [math.CO], 2017.
%H A110501 Ange Bigeni, <a href="https://arxiv.org/abs/1712.01929">Combinatorial interpretations of the Kreweras triangle in terms of subset tuples</a>, arXiv:1712.01929 [math.CO], 2017.
%H A110501 Ange Bigeni, <a href="https://doi.org/10.1016/j.jcta.2018.08.005">A generalization of the Kreweras triangle through the universal sl_2 weight system</a>, Journal of Combinatorial Theory, Series A (2019) Vol. 161, 309-326.
%H A110501 A. Burstein, M. Josuat-Vergès and W. Stromquist, <a href="https://www.researchgate.net/publication/228671145_New_Dumont_permutations">New Dumont permutations</a>, Pure Math. Appl. (Pu.M.A.) 21 (2010), no. 2, 177-206.
%H A110501 E. Clark and R. Ehrenborg, <a href="http://dx.doi.org/10.1016/j.disc.2013.03.014">The excedance algebra</a>, Discr. Math., 313 (2013), 1429-1435.
%H A110501 Bishal Deb and Alan D. Sokal, <a href="https://arxiv.org/abs/2212.07232">Classical continued fractions for some multivariate polynomials generalizing the Genocchi and median Genocchi numbers</a>, arXiv:2212.07232 [math.CO], 2022. See pp. 4, 12.
%H A110501 Bishal Deb, <a href="https://arxiv.org/abs/2304.14487">Continued fractions using a Laguerre digraph interpretation of the Foata-Zeilberger bijection and its variants</a>, arXiv:2304.14487 [math.CO], 2023. See pp. 4, 11.
%H A110501 D. Dumont, <a href="http://dx.doi.org/10.1016/0012-365X(72)90039-8">Sur une conjecture de Gandhi concernant les nombres de Genocchi</a>, Discrete Mathematics 1 (1972) 321-327.
%H A110501 D. Dumont, <a href="http://dx.doi.org/10.1215/S0012-7094-74-04134-9">Interpretations combinatoires des nombres de Genocchi</a>, Duke Math. J., 41 (1974), 305-318.
%H A110501 Dominique Dumont and Dominique Foata, <a href="http://www.numdam.org/item?id=BSMF_1976__104__433_0">Une propriété de symétrie des nombres de Genocchi</a> Bull. Soc. Math. France 104 (1976), no. 4, 433-451. MR0434830 (55 #7794)
%H A110501 D. Dumont and G. Viennot, <a href="/A110501/a110501.pdf"> A combinatorial interpretation of the Seidel generation of Genocchi numbers</a>, Preprint, Annotated scanned copy.
%H A110501 Dominique Dumont and Gérard Viennot, <a href="http://dx.doi.org/10.1016/S0167-5060(08)70696-4">A combinatorial interpretation of the Seidel generation of Genocchi numbers</a>, 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).
%H A110501 A. L. Edmonds and S, Klee, <a href="http://arxiv.org/abs/1210.7396">The combinatorics of hyperbolized manifolds</a>, arXiv:1210.7396 [math.CO], 2012.
%H A110501 Richard Ehrenborg and Einar Steingrímsson, <a href="http://dx.doi.org/10.1006/eujc.1999.0370">Yet another triangle for the Genocchi numbers</a>, European J. Combin. 21 (2000), no. 5, 593-600. MR1771988 (2001h:05008)
%H A110501 Sen-Peng Eu, Tung-Shan Fu, Hsin-Hao Lai, and Yuan-Hsun Lo, <a href="https://arxiv.org/abs/2103.09130">Gamma-positivity for a Refinement of Median Genocchi Numbers</a>, arXiv:2103.09130 [math.CO], 2021.
%H A110501 Vincent Froese and Malte Renken, <a href="https://arxiv.org/abs/2210.16281">Terrain-like Graphs and the Median Genocchi Numbers</a>, arXiv:2210.16281 [math.CO], 2022.
%H A110501 J. M. Gandhi, <a href="http://www.jstor.org/stable/2317385">Research Problems: A Conjectured Representation of Genocchi Numbers</a>, Amer. Math. Monthly 77 (1970), no. 5, 505-506. MR1535914
%H A110501 Ira M. Gessel, <a href="https://doi.org/10.5281/zenodo.7625111">On the Almkvist-Meurman Theorem for Bernoulli Polynomials</a>, Integers (2023) Vol. 23, #A14.
%H A110501 S. W. Golomb and E. C. Posner, <a href="http://dx.doi.org/10.1109/TIT.1964.1053680">Rook Domains, Latin Squares, Affine Planes, and Error-Distributing Codes</a>, Transactions of the Information Theory Group of the IEEE, Vol. 10, No. 3 (1964), 196-208.
%H A110501 Guo-Niu Han, <a href="http://www-irma.u-strasbg.fr/~guoniu/papers/p77puzzle.pdf">Enumeration of Standard Puzzles</a>
%H A110501 Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a> [Cached copy]
%H A110501 Guo-Niu Han and Jing-Yi Liu, <a href="https://doi.org/10.1016/j.ejc.2018.02.041">Combinatorial proofs of some properties of tangent and Genocchi numbers</a>, European Journal of Combinatorics, Vol. 71 (2018), pp. 99-110; <a href="https://arxiv.org/abs/1707.08882">arXiv preprint</a>, arXiv:1707.08882 [math.CO], 2017-2018.
%H A110501 Florent Hivert and Olivier Mallet, <a href="https://doi.org/10.46298/dmtcs.2928">Combinatorics of k-shapes and Genocchi numbers</a>, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 493-504.
%H A110501 Alexander Lazar and Michelle L. Wachs, <a href="https://arxiv.org/abs/1910.07651">The Homogenized Linial Arrangement and Genocchi Numbers</a>, arXiv:1910.07651 [math.CO], 2019.
%H A110501 Zhicong Lin and Sherry H.F. Yan, <a href="https://arxiv.org/abs/2108.03790">Cycles on a multiset with only even-odd drops</a>, arXiv:2108.03790 [math.CO], 2021. See also <a href="https://doi.org/10.1016/j.disc.2021.112683">Disc. Math.</a> (2022) Vol. 345, No. 2, 112683.
%H A110501 A. H. Morales and D. G. Zhu, <a href="https://arxiv.org/abs/2007.05006">On the Okounkov--Olshanski formula for standard tableaux of skew shape</a>, arXiv:2007.05006 [math.CO], 2020.
%H A110501 Qiongqiong Pan and Jiang Zeng, <a href="https://arxiv.org/abs/2108.03200">Cycles of even-odd drop permutations and continued fractions of Genocchi numbers</a>, arXiv:2108.03200 [math.CO], 2021.
%H A110501 John Riordan and Paul R. Stein, <a href="http://dx.doi.org/10.1016/0012-365X(73)90131-3">Proof of a conjecture on Genocchi numbers</a>, Discrete Math. 5 (1973), 381-388. MR0316372 (47 #4919) - From _N. J. A. Sloane_, Jun 12 2012
%H A110501 Alan Sokal, <a href="/A110501/a110501.txt">Table of n, a(n) for n = 1..10000 [315 MB file]</a>
%H A110501 H. M. Terrill and E. M. Terrill, <a href="https://ur.booksc.eu/ireader/2106189">Tables of numbers related to the tangent coefficients</a>, J. Franklin Inst., 239 (1945), 66-67.
%F A110501 (-1)^n * a(n) = A036968(2*n) = A001469(n).
%F A110501 a(n) = 2*(-1)^n*(1-4^n)*B_{2*n} (B = A027641/A027642 are Bernoulli numbers).
%F A110501 A002105(n) = 2^(n-1)/n * a(n). - _Don Knuth_, Jan 16 2007
%F A110501 A000111(2*n-1) = a(n)*2^(2*n-2)/n. - _Alejandro H. Morales_, Jul 26 2020
%F A110501 E.g.f.: x * tan(x/2) = Sum_{k > 0} a(k) * x^(2*k) / (2*k)!.
%F A110501 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
%F A110501 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
%F A110501 a(n) = Sum_{k = 0..2*n} (-1)^(n-k+1)*Stirling2(2*n, k)*A059371(k). - _Vladeta Jovovic_, Feb 07 2004
%F A110501 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
%F A110501 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
%F A110501 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
%F A110501 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
%F A110501 a(n+1) = Sum_{k=0..n} A211183(n, k)*2^(n-k). - _Philippe Deléham_, Feb 03 2013
%F A110501 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
%F A110501 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
%F A110501 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
%F A110501 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
%F A110501 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
%F A110501 a(n) = 2*n*|euler(2*n-1, 0)|. - _Peter Luschny_, Jun 09 2016
%F A110501 a(n) = 4^(1-n) * (4^n-1) * Pi^(-2*n) * (2*n)! * zeta(2*n). - _Daniel Suteu_, Oct 14 2016
%F A110501 a(n) ~ 8*Pi*(2^(2*n)-1)*(n/(Pi*exp(1)))^(2*n+1/2)*exp(1/2+(1/24)/n-(1/2880)/n^3+(1/40320)/n^5+...). [Given in A001469 by _Peter Luschny_, Jul 24 2013, copied May 14 2022.]
%F A110501 a(n) = A000182(n) * n / 4^(n-1) (Han and Liu, 2018). - _Amiram Eldar_, May 17 2024
%e A110501 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! + ...
%e A110501 O.g.f.: A(x) = x + x^2 + 3*x^3 + 17*x^4 + 155*x^5 + 2073*x^6 + ...
%e A110501 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
%e A110501 From _Gary W. Adamson_, Jul 19 2011: (Start)
%e A110501 The first few rows of production matrix M are:
%e A110501 1, 2, 0, 0, 0, 0, ...
%e A110501 1, 3, 3, 0, 0, 0, ...
%e A110501 1, 4, 6, 4, 0, 0, ...
%e A110501 1, 5, 10, 10, 5, 0, ...
%e A110501 1, 6, 15, 20, 15, 6, ... (End)
%p A110501 A110501 := proc(n)
%p A110501 2*(-1)^n*(1-4^n)*bernoulli(2*n) ;
%p A110501 end proc:
%p A110501 seq(A110501(n),n=0..10) ; # _R. J. Mathar_, Aug 02 2013
%t A110501 a[n_] := 2*(4^n - 1) * BernoulliB[2n] // Abs; Table[a[n], {n, 19}] (* _Jean-François Alcover_, May 23 2013 *)
%o A110501 (PARI) {a(n) = if( n<1, 0, 2 * (-1)^n * (1 - 4^n) * bernfrac( 2*n))};
%o A110501 (PARI) {a(n) = if( n<1, 0, (2*n)! * polcoeff( x * tan(x/2 + x * O(x^(2*n))), 2*n))};
%o A110501 (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 */
%o A110501 (PARI) upto(n) = my(v1, v2, v3); v1 = vector(n, i, 0); v1[1] = 1; v2 = vector(n-1, i, ((i+1)^2)\4); v3 = v1; for(i=2, n, for(j=2, i-1, v1[j] += v2[i-j+1]*v1[j-1]); v1[i] = v1[i-1]; v3[i] = v1[i]); v3 \\ _Mikhail Kurkov_, Aug 28 2025
%o A110501 (Sage) # Algorithm of L. Seidel (1877)
%o A110501 # n -> [a(1), ..., a(n)] for n >= 1.
%o A110501 def A110501_list(n) :
%o A110501 D = []; [D.append(0) for i in (0..n+2)]; D[1] = 1
%o A110501 R = [] ; b = True
%o A110501 for i in(0..2*n-1) :
%o A110501 h = i//2 + 1
%o A110501 if b :
%o A110501 for k in range(h-1,0,-1) : D[k] += D[k+1]
%o A110501 else :
%o A110501 for k in range(1,h+1,1) : D[k] += D[k-1]
%o A110501 b = not b
%o A110501 if b : R.append(D[h])
%o A110501 return R
%o A110501 A110501_list(19) # _Peter Luschny_, Apr 01 2012
%o A110501 (Sage) [2*(-1)^n*(1-4^n)*bernoulli(2*n) for n in (1..20)] # _G. C. Greubel_, Nov 28 2018
%o A110501 (Magma) [Abs(2*(4^n-1)*Bernoulli(2*n)): n in [1..20]]; // _Vincenzo Librandi_, Jul 28 2017
%o A110501 (Python)
%o A110501 from sympy import bernoulli
%o A110501 def A110501(n): return ((2<<(m:=n<<1))-2)*abs(bernoulli(m)) # _Chai Wah Wu_, Apr 14 2023
%Y A110501 Cf. A000182, A001469, A002105, A005439, A036968, A211183, A297703.
%K A110501 nonn,changed
%O A110501 1,3
%A A110501 _Michael Somos_, Jul 23 2005
%E A110501 Edited by _M. F. Hasler_, Mar 22 2015