A000366 -id:A000366 - cp's OEIS frontend

A130168 a(n) = (b(n) + b(n+1))/3, where b(n) = A000366(n).

1, 3, 15, 111, 1131, 15123, 256335, 5364471, 135751731, 4084163643, 144039790455, 5884504366431, 275643776229531, 14673941326078563, 880908054392169375, 59226468571935857991, 4432461082611507366531, 367227420727722013775883, 33514867695588319595233095

Offset: 2

Don Knuth, Aug 02 2007

nonn

As remarked by Gessel, A000366 has a combinatorial interpretation via a certain 2n X n array; this sequence is for a similar array of size (2n-1) X (n-1).

In effect, Dellac gives a combinatorial reason why the elements of A000366 are alternately -1 and +1 modulo 3. Dellac also shows that all the terms of this sequence are odd.

Hippolyte Dellac, Note sur l'élimination, méthode de parallélogramme, Annales de la Faculté des Sciences de Marseille, XI (1901), 141-164. [Warning 76 Mb; go to p. 81 in the pdf file]

Cf. A000366, A130169.

b[n_] := (-2^(-1))^(n-2)*Sum[Binomial[n, k]*(1-2^(n+k+1))* BernoulliB[n+k+1], {k, 0, n}];
a[n_] := (b[n] + b[n+1])/3;
a /@ Range[2, 20] (* Jean-François Alcover, Apr 08 2021 *)

from math import comb
from sympy import bernoulli
def A130168(n): return (abs((2-(2<>n-1)//3 # Chai Wah Wu, Apr 14 2023

G.f.: 2*(1+x)/(3*x^3)*Q(0) - 2/(3*x) - 1/x^2 - 2/(3*x^3), where Q(k) = 1 - x*(k+1)^2/( x*(k+1)^2 - 2/(1 - x*(k+1)^2/( x*(k+1)^2 - 2/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Oct 22 2013

a(n) = |(2-2^(n+2))*Bernoulli(n+1) - (n+1)*(1-2^(2n+2))*Bernoulli(2n+2) - (1-2^(2n+3))*Bernoulli(2n+3) + Sum_{k=0..n-1} (2*binomial(n,k+1)-binomial(n+1,k))*(1-2^(n+k+2))*Bernoulli(n+k+2)|/(3*2^(n-1)). - Chai Wah Wu, Apr 14 2023

A005439 Genocchi medians (or Genocchi numbers of second kind).

Original entry on oeis.org

1, 1, 2, 8, 56, 608, 9440, 198272, 5410688, 186043904, 7867739648, 401293838336, 24290513745920, 1721379917619200, 141174819474169856, 13266093250285568000, 1415974941618255921152, 170361620874699124637696, 22948071824232932086513664, 3439933090471867097102680064

Offset: 0

Simon Plouffe

a(n) is the number of Boolean functions of n variables whose ROBDD (reduced ordered binary decision diagram) contains exactly n branch nodes, one for each variable. - Don Knuth, Jul 11 2007
The earliest known reference for these numbers is Seidel (1877, pages 185 and 186). - Don Knuth, Jul 13 2007
Hankel transform of 1,1,2,8,... is A168488. - Paul Barry, Nov 27 2009
According to Hetyei [2017], alternation acyclic tournaments "are counted by the median Genocchi numbers"; an alternation acyclic tournament "does not contain a cycle in which descents and ascents alternate." - Danny Rorabaugh, Apr 25 2017
The n-th Genocchi number of the second kind is also the number of collapsed permutations in (2n) letters. A permutation pi of size 2n is said to be collapsed if 1+floor(k/2) <= pi^{-1}(k) <= n + floor(k/2). There are 2 collapsed permutations of size 4, namely 1234 and 1324. - Arvind Ayyer, Oct 23 2020
For any positive integer n, a(n) is (-1)^n times the permanent of the 2n X 2n matrix M with M(j, k) = floor((2*j-k-1)/(2*n)). This former conjecture of Luschny, inspired by a conjecture of Zhi-Wei Sun in A036968, was proven by Fu, Lin and Sun (see link). - Peter Luschny, Sep 07 2021 [updated Sep 24 2021]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Muniru A Asiru, Table of n, a(n) for n = 0..270 (terms n = 1..100 from T. D. Noe)
A. Ayyer, D. Hathcock, and P. Tetali, Toppleable Permutations, Excedances and Acyclic Orientations, arXiv:2010.11236 [math.CO], 2020.
Paul Barry, Series reversion with Jacobi and Thron continued fractions, arXiv:2107.14278 [math.NT], 2021.
Beáta Bényi, A Bijection for the Boolean Numbers of Ferrers Graphs, Graphs and Combinatorics (2022) Vol. 38, No. 10.
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.
Ange Bigeni, A generalization of the Kreweras triangle through the universal sl_2 weight system, Journal of Combinatorial Theory, Series A (2019) Vol. 161, 309-326.
Alin Bostan, Lucia Di Vizio, and Kilian Raschel, Singular walks in the quarter plane and Bernoulli numbers, arXiv:2504.13542 [math.CO], 2025. See p. 28.
Alexander Burstein, Sergi Elizalde, and Toufik Mansour, Restricted Dumont permutations, Dyck paths and noncrossing partitions, arXiv:math/0610234 [math.CO], 2006. [Theorem 3.5]
Kwang-Wu Chen, An Interesting Lemma for Regular C-fractions, J. Integer Seqs., Vol. 6, 2003.
Shane Chern, Parity considerations for drops in cycles on {1,2,...,n}, arXiv:2112.02074 [math.CO], 2021.
Bishal Deb and Alan D. Sokal, Classical continued fractions for some multivariate polynomials generalizing the Genocchi and median Genocchi numbers, arXiv:2212.07232 [math.CO], 2022. See pp. 14-15.
Bishal Deb, Continued fractions using a Laguerre digraph interpretation of the Foata-Zeilberger bijection and its variants, arXiv:2304.14487 [math.CO], 2023. See p. 4.
Bishal Deb, Cyclic sieving phenomena via combinatorics of continued fractions, arXiv:2508.13709 [math.CO], 2025. See p. 38.
D. Dumont and J. Zeng, Polynômes d'Euler et les fractions continues de Stieltjes-Rogers, Ramanujan J. 2 (1998) 3, 387-410.
Richard Ehrenborg and Einar Steingrímsson, Yet another triangle for the Genocchi numbers, European J. Combin. 21 (2000), no. 5, 593-600. MR1771988 (2001h:05008).
Sen-Peng Eu, Tung-Shan Fu, Hsin-Hao Lai, and Yuan-Hsun Lo, Gamma-positivity for a Refinement of Median Genocchi Numbers, arXiv:2103.09130 [math.CO], 2021.
Vincent Froese and Malte Renken, Terrain-like Graphs and the Median Genocchi Numbers, arXiv:2210.16281 [math.CO], 2022.
Shishuo Fu, Zhicong Lin, and Zhi-Wei Sun, Proofs of five conjectures relating permanents to combinatorial sequences, arXiv:2109.11506 [math.CO], 2021.
Shishuo Fu, Zhicong Lin, and Zhi-Wei Sun, Permanent identities, combinatorial sequences, and permutation statistics, Advances in Applied Mathematics, Volume 163, Part A, 102789 (2025).
I. M. Gessel, Applications of the classical umbral calculus, arXiv:math/0108121 [math.CO], 2001.
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.
Gábor Hetyei, Alternation acyclic tournaments, arXiv:math/1704.07245 [math.CO], 2017.
G. Kreweras, Sur les permutations comptées par les nombres de Genocchi de 1-ière et 2-ième espèce, Europ. J. Comb., vol. 18, pp. 49-58, 1997. (See also page 76.)
Alexander Lazar and Michelle L. Wachs, The Homogenized Linial Arrangement and Genocchi Numbers, arXiv:1910.07651 [math.CO], 2019.
Qiongqiong Pan and Jiang Zeng, Cycles of even-odd drop permutations and continued fractions of Genocchi numbers, arXiv:2108.03200 [math.CO], 2021.
A. Randrianarivony and J. Zeng, Une famille de polynômes qui interpole plusieurs suites classiques de nombres, Adv. Appl. Math. 17 (1996), 1-26. In French.
L. Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187.
Alan Sokal, Table of n, a(n) for n = 1..10000 [315 MB file]
Zhi-Wei Sun, Arithmetic properties of some permanents, arXiv:2108.07723 [math.GM], 2021.
G. Viennot, Interprétations combinatoires des nombres d'Euler et de Genocchi, Seminar on Number Theory, 1981/1982, Exp. No. 11, 94 pp., Univ. Bordeaux I, Talence, 1982.

Cf. A000366, A036968, A110501, A297703.

List([1..20],n->2*(-1)^n*Sum([0..n],k->Binomial(n,k)*(1-2^(n+k+1))*Bernoulli(n+k+1))); # Muniru A Asiru, Nov 29 2018

[2*(-1)^n*(&+[Binomial(n, k)*(1-2^(n+k+1))*Bernoulli(n+k+1): k in [0..n]]): n in [1..20]]; // G. C. Greubel, Nov 28 2018

seq(2*(-1)^n*add(binomial(n,k)*(1 - 2^(n+k+1))*bernoulli(n+k+1), k=0..n), n=0..20); # G. C. Greubel, Oct 18 2019

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,16}] (* Jean-François Alcover, Jul 18 2011, after PARI prog. *)

a(n)=2*(-1)^n*sum(k=0,n,binomial(n,k)*(1-2^(n+k+1))* bernfrac(n+k+1))

a(n)=local(CF=1+x*O(x^(n+2)));if(n<0,return(0), for(k=1,n+1,CF=1/(1-((n-k+1)\2+1)^2*x*CF));return(Vec(CF)[n+2])) \\ Paul D. Hanna

from math import comb
from sympy import bernoulli
def A005439(n): return (-2 if n&1 else 2)*sum(comb(n,k)*(1-(1<Chai Wah Wu, Apr 14 2023

# Algorithm of L. Seidel (1877)
# n -> [a(1), ..., a(n)] for n >= 1.
def A005439_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]
        if b : R.append(D[1])
        b = not b
    return R
A005439_list(18) # Peter Luschny, Apr 01 2012

[2*(-1)^n*sum(binomial(n,k)*(1-2^(n+k+1))*bernoulli(n+k+1) for k in (0..n)) for n in (1..20)] # G. C. Greubel, Oct 18 2019

a(n) = T(n, 1) where T(1, x) = 1; T(n, x) = (x+1)*((x+1)*T(n-1, x+1)-x*T(n-1, x)); see A058942.
a(n) = A000366(n)*2^(n-1).
a(n) = 2 * (-1)^n * Sum_{k=0..n} binomial(n, k)*(1-2^(n+k+1))*B(n+k+1), with B(n) the Bernoulli numbers. - Ralf Stephan, Apr 17 2004
O.g.f.: 1 + x*A(x) = 1/(1-x/(1-x/(1-4*x/(1-4*x/(1-9*x/(1-9*x/(... -[(n+1)/2]^2*x/(1-...)))))))) (continued fraction). - Paul D. Hanna, Oct 07 2005
G.f.: (of 1,1,2,8,...) 1/(1-x-x^2/(1-5*x-16*x^2/(1-13*x-81*x^2/(1-25*x-256*x^2/(1-41*x-625*x^2/(1-... (continued fraction). - Paul Barry, Nov 27 2009
O.g.f.: Sum_{n>=0} n!*(n+1)! * x^(n+1) / Product_{k=1..n} (1 + k*(k+1)*x). - Paul D. Hanna, May 10 2012
From Sergei N. Gladkovskii, Dec 14 2011, Dec 27 2012, May 29 2013, Oct 09 2013, Oct 24 2013, Oct 27 2013: (Start)
Continued fractions:
G.f.: A(x) = 1/S(0), S(k) = 1 - x*(k+1)*(k+2)/(1 - x*(k+1)*(k+2)/S(k+1)).
G.f.: A(x) = -1/S(0), S(k) = 2*x*(k+1)^2 - 1 - x^2*(k+1)^2*(k+2)^2/S(k+1).
G.f.: A(x) = 1/G(0) where G(k) = 1 - x*(k+1)^2/(1 - x*(k+1)^2/G(k+1)).
G.f.: 2/G(0), where G(k) = 1 + 1/(1 - 1/(1 - 1/(4*x*(k+1)) + 1/G(k+1))).
G.f.: Q(0)/x - 1/x, where Q(k) = 1 - x*(k+1)^2/( x*(k+1)^2 - 1/(1 - x*(k+1)^2/( x*(k+1)^2 - 1/Q(k+1)))).
G.f.: T(0)/(1-2*x), where T(k) = 1 - x^2*((k + 2)*(k+1))^2/(x^2*((k + 2)*(k+1))^2 - (1 - 2*x*k^2 - 4*x*k - 2*x)*(1 - 2*x*k^2 - 8*x*k - 8*x)/T(k+1)).
G.f.: R(0), where R(k) = 1 - x*(k+1)*(k+2)/( x*(k+1)*(k+2) - 1/(1 - x*(k+1)*(k+2)/( x*(k+1)*(k+2) - 1/R(k+1) ))). (End)
a(n) ~ 2^(2*n+4) * n^(2*n+3/2) / (exp(2*n) * Pi^(2*n+1/2)). - Vaclav Kotesovec, Oct 28 2014
Rewriting the above: a(n) ~ 4*(2*n+1)! / Pi^(2*n+1). Compare to Genocchi numbers A110501(n) = g_n ~ 4*(2*n)! / Pi^(2*n). So these are indeed like "Genocchi medians" g_{n + 1/2}. - Alan Sokal, May 13 2022
Asymptotic expansion:  a(n) ~ 4*(2*n+1)! * Pi^(-(2*n+1)) * (1 + (Pi^2/16)/n + (Pi^2 (Pi^2 - 16)/512)/n^2 + (Pi^2 (Pi^4 + 384)/24576)/n^3 + (Pi^2 (Pi^6 + 96*Pi^4 + 768*Pi^2 - 12288)/1572864)/n^4 + (Pi^2 (Pi^8 + 320*Pi^6 + 12800*Pi^4 + 491520)/125829120)/n^5 + ...) --- Proof uses binomial sum for Genocchi medians in terms of Genocchi or Bernoulli numbers, combined with leading term of convergent sum (with exponentially small corrections) for the latter.  Can also check against the 10000 term a-file. - Alan Sokal, May 23 2022.
a(n) = n!^2 * [x^n*y^n] exp(x)*f(x-y), where f(x) is the derivative of the Genocchi number generating function 2*x/(exp(x)+1). - Ira M. Gessel, Jul 23 2024

More terms and additional comments from David W. Wilson, Jan 11 2001

a(0)=1 prepended by Peter Luschny, Apr 14 2023

A211183 Triangle T(n,k), 0<=k<=n, read by rows, given by (0, 1, 1, 3, 3, 6, 6, 10, 10, 15, ...) DELTA (1, 0, 2, 0, 3, 0, 4, 0, 5, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 4, 1, 0, 7, 19, 11, 1, 0, 38, 123, 107, 26, 1, 0, 295, 1076, 1195, 474, 57, 1, 0, 3098, 12350, 16198, 8668, 1836, 120, 1, 0, 42271, 180729, 268015, 176091, 52831, 6549, 247, 1, 0, 726734, 3290353, 5369639, 4105015, 1564817, 287473, 22145

Offset: 0

Philippe Deléham, Feb 02 2013

			Triangle begins :
1;
0, 1;
0, 1, 1;
0, 2, 4, 1;
0, 7, 19, 11, 1;
0, 38, 123, 107, 26, 1;
0, 295, 1076, 1195, 474, 57, 1;
0, 3098, 12350, 16198, 8668, 1836, 120, 1;
0, 42271, 180729, 268015, 176091, 52831, 6549, 247, 1;
0, 726734, 3290353, 5369639, 4105015, 1564817, 287473, 22145, 502, 1; ...

Paul D. Hanna, Rows n = 0..31, flattened.

Cf. A000366, A110501, A211194, A221972.

T(n,k)=polcoeff(polcoeff(sum(m=0, n, m!*x^m*prod(k=1, m, (y + (k-1)/2)/(1+(k*y+k*(k-1)/2)*x+x*O(x^n)))), n,x),k,y)
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print()) \\ Paul D. Hanna, Feb 03 2013

Sum_{k, 0<=k<=n}T(n,k)*x^(n-k) = A000012(n), A000366(n+1), A110501(n+1), A211194(n), A221972(n) for x = 0, 1, 2, 3, 4 respectively.
T(n,n-1) = A000295(n).
T(n,1) = A000366(n).
G.f.: A(x,y) = Sum_{n>=0} n! * x^n * Product_{k=1..n} (y + (k-1)/2) / (1 + (k*y + k*(k-1)/2)*x). - Paul D. Hanna, Feb 03 2013

A014784 Triangle of numbers associated with Genocchi numbers.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 7, 12, 12, 7, 38, 69, 81, 69, 38, 295, 552, 702, 702, 552, 295, 3098, 5901, 7857, 8559, 7857, 5901, 3098, 42271, 81444, 111618, 128034, 128034, 111618, 81444, 42271, 726734, 1411197, 1971945, 2339631, 2467665, 2339631, 1971945

Offset: 0

N. J. A. Sloane

G. Kreweras, Sur les permutations comptées par les nombres de Genocchi de 1-ière et 2-ième espèce, Europ. J. Comb., vol. 18, pp. 49-58, 1997.
G. Kreweras and D. Dumont, Sur les anagrammes alternés, Discrete Mathematics, Volume 211, Issues 1-3, 28 January 2000, Pages 103-110. Zbl 0944.05003.

Cf. A035003.

Columns include A000366 and A102078. Row sums are in A000366.

a[ 1, 1 ]=1; a[ n_, k_ ] := 0 /; (k>n || k<=0); a[ n_, 1 ]=Sum[ a[ n-1, i ], {i, 1, n-1} ] a[ n_, k_ ] := a[ n, k ]=2a[ n, k-1 ]-a[ n, k-2 ]-a[ n-1, k-1 ]-a[ n-1, k-2 ]; Flatten[ Table[ a[ n, i ], {n, 1, 10}, {i, 1, n} ] ]

More terms from Erich Friedman.

A218826 Number of indecomposable (by concatenation) alternating n-anagrams.

Original entry on oeis.org

1, 1, 4, 25, 217, 2470, 35647, 637129, 13843948, 360022957, 11054457253, 396003680518, 16377463914091, 774714094061221, 41572230979229284, 2512149910125036865, 169831839578092130017, 12769241823369505582150, 1062122471116082751430087, 97264621940872013476357969

Offset: 1

Michel Marcus, Nov 07 2012

nonn

Alternating anagrams enumeration is related to A000366 by a(n) = A000366(n+1).

For all n, a(n) are periodically congruent to 1, 1 and 4 modulo 6.

Andrew Howroyd, Table of n, a(n) for n = 1..200
G. Kreweras and J. Barraud, Anagrammes alternés, European Journal of Combinatorics,Volume 18, Issue 8, November 1997, Pages 887-891.
G. Kreweras and D. Dumont, Sur les anagrammes alternés, Discrete Mathematics, Volume 211, Issues 1-3, 28 January 2000, Pages 103-110.

Cf. A000366, A218827. First column of A239894.

a000366(n)= {return((-1/2)^(n-2)*sum(k=0, n, binomial(n, k)*(1-2^(n+k+1))*bernfrac(n+k+1)));}
bi(n,k) = {if (matb[n,k] == 0, if (n==k, v=1, if (k==1, v = b(n),v = sum(i=1, n-k+1, b(i)*bi(n-i,k-1)););); matb[n,k] = v;); return (matb[n,k]);}
b(n) = {if (n==1, return(a000366(n+1)), return(a000366(n+1) - sum(i=2, n, bi(n,i))));}
allb(m) = {matb = matrix(m,m); for (i=1, m, print1(b(i), ", "););}

G.f.: (1-Q(0))/x, where Q(k) = 1 - ((k+1)*(k+2)/2)*x/(1 - ((k+1)*(k+2)/2)*x/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 03 2013
G.f.: 1/Q(0), where Q(k) = 1 - ((k+1)*(k+2)/2)*x/(1 - ((k+2)*(k+3)/2)*x/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Nov 17 2013
a(n) = A000366(n + 1) - Sum_{k=2..n} A239894(n, k). - Andrew Howroyd, Feb 25 2020

A239894 Triangle read by rows: T(n,k) (n>=1, 1 <= k <= n) = number of alternating anagrams on n letters (of length 2n) which are decomposable into at most k slices.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 25, 9, 3, 1, 217, 58, 15, 4, 1, 2470, 500, 100, 22, 5, 1, 35647, 5574, 861, 152, 30, 6, 1, 637129, 78595, 9435, 1313, 215, 39, 7, 1, 13843948, 1376162, 130159, 14192, 1870, 290, 49, 8, 1, 360022957, 29417919, 2232792, 191850, 20001, 2547, 378, 60, 9, 1

Offset: 1

N. J. A. Sloane, Apr 04 2014

			Triangle begins:
1
1 1
4 2 1
25 9 3 1
217 58 15 4 1
...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
Kreweras, G.; Dumont, D. , Sur les anagrammes alternés. (French) [On alternating anagrams] Discrete Math. 211 (2000), no. 1-3, 103--110. MR1735352 (2000h:05013).

Row sums are A000366.

First column is A218826.

\\ here G(n) is A000366(n).
G(n)={(-1/2)^(n-2)*sum(k=0, n, binomial(n, k)*(1-2^(n+k+1))*bernfrac(n+k+1))}
A(n)={my(M=matrix(n,n)); for(n=1, n, for(k=2, n, M[n,k] = sum(i=1, n-k+1, M[i,1]*M[n-i, k-1])); M[n,1]=G(n+1)-sum(i=2, n, M[n,i])); M}
{my(T=A(10)); for(n=1, #T, print(T[n, 1..n]))} \\ Andrew Howroyd, Feb 24 2020

T(n,k) = b(1)*T(n-1,k-1)+b(2)*T(n-2,k-1)+...+b(n-k+1)*T(k-1,k-1), where b(i) = A218826(i) for k > 1.

T(n,k) = Sum_{i=1..n-k+1} A218826(i)*T(n-i, k-1) for k > 1. - Andrew Howroyd, Feb 24 2020

Terms a(16) and beyond from Andrew Howroyd, Feb 19 2020

A098278 D(n,0)/2^n, where D(n,x) is triangle A098277.

Original entry on oeis.org

1, 1, 3, 21, 267, 5349, 154923, 6120741, 316271787, 20701782309, 1673934058923, 163850823271461, 19093313058395307, 2611858473935397669, 414452507370456337323, 75508557963926980473381

Offset: 0

Ralf Stephan, Sep 07 2004

nonn

This is related to formula (1.7) in Lazar and Wachs reference.

Apparently all terms (except the initial 1s) have 3-valuation 1. - F. Chapoton, Jul 31 2021

			G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 267*x^4 + 5349*x^5 + ...
where A(x) = 1 + x/(1+x) + 2!^2*x^2/((1+x)*(1+3*x)) + 3!^2*x^3/((1+x)*(1+3*x)*(1+6*x)) + 4!^2*x^4/((1+x)*(1+3*x)*(1+6*x)*(1+10*x)) + ... - _Paul D. Hanna_, Sep 05 2012

Ange Bigeni and Evgeny Feigin, Symmetric Dellac configurations, arXiv:1808.04275 [math.CO], 2018.
Alexander Lazar and Michelle L. Wachs, On the homogenized Linial arrangement: intersection lattice and Genocchi numbers, Séminaire Lotharingien de Combinatoire, 82B.93 (FPSAC 2019).
A. Randrianarivony and J. Zeng, Une famille de polynômes qui interpole plusieurs suites..., Adv. Appl. Math. 17 (1996), 1-26.

Cf. A000366.

d[0, ] = 1; d[n, x_] := d[n, x] = (x+1)(x+2)d[n-1, x+2]-x(x+1)d[n-1, x];
a[n_] := d[n, 0]/2^n;
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Oct 26 2018 *)

{a(n)=polcoeff(sum(m=0, n, m!^2*x^m/prod(k=1, m, 1+k*(k+1)/2*x +x*O(x^n))), n)} \\ Paul D. Hanna, Sep 05 2012

G.f.: Sum_{n>=0} a(n)*x^n = 1/(1-1*1*x/(1-1*2*x/(1-2*3*x/(1-2*4*x/...)))).
G.f.: Sum_{n>=0} n!^2 * x^n / Product_{k=1..n} (1 + k*(k+1)/2*x). - Paul D. Hanna, Sep 05 2012
G.f.: 1/G(0) where G(k) = 1 - x*(k+1)*(2*k+1)/(1 - x*(k+1)*(2*k+2)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 14 2013.
a(n+1) = Sum_{k=0..n} A098277(n,k)*(1/2)^k. - Philippe Deléham, Feb 08 2013

A130169 a(1) = 1; for n>1, a(n) = (c(n) + c(n+1))/2, where c(n) = A130168(n).

Original entry on oeis.org

1, 2, 9, 63, 621, 8127, 135729, 2810403, 70558101, 2109957687, 74061977049, 3014272078443, 140764140297981, 7474792551154047, 447790997859123969, 30053688313164013683, 2245843775591721612261, 185829940905166760571207, 16941047558158020804504489

Offset: 1

Don Knuth, Aug 02 2007

nonn

Hippolyte Dellac, Note sur l'élimination, méthode de parallélogramme, Annales de la Faculté des Sciences de Marseille, XI (1901), 141-164. [Warning 76 Mb; go to p. 81 in the pdf file]

Cf. A000366, A130168.

b[n_] := (-2^(-1))^(n-2)*Sum[Binomial[n, k]*(1-2^(n+k+1))* BernoulliB[n+k+1], {k, 0, n}];
c[n_] := (b[n] + b[n+1])/3;
a[n_] := If[n == 1, 1, (c[n] + c[n+1])/2];
a /@ Range[1, 19] (* Jean-François Alcover, Apr 08 2021 *)

G.f.: (1+x)^2/(3*x^3)*Q(0) + (x^3 - 5*x^2 - 5*x - 2)/(6*x^3), where Q(k) = 1 - x*(k+1)^2/( x*(k+1)^2 - 2/(1 - x*(k+1)^2/( x*(k+1)^2 - 2/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Oct 22 2013

A218827 Number of indecomposable (by shuffling) alternating n-anagrams.

Original entry on oeis.org

1, 1, 3, 16, 129, 1438, 20955, 384226, 8623101, 231978454, 7359117591, 271673905642, 11543742745689, 559348370431630, 30659822500574739, 1887796293833267746, 129757032076160998677, 9900820197631733600518, 834421415151529202479023, 77318409826165250051727514

Offset: 1

Michel Marcus, Nov 07 2012

nonn

Alternating anagrams enumeration is related to A000366 by a(n) = A000366(n+1).

For n>2, a(n) are alternatively congruent to 3 and -2 modulo 18.

G. Kreweras and J. Barraud, Anagrammes alternés, European Journal of Combinatorics,Volume 18, Issue 8, November 1997, Pages 887-891.
G. Kreweras and D. Dumont, Sur les anagrammes alternés, Discrete Mathematics, Volume 211, Issues 1-3, 28 January 2000, Pages 103-110.

Cf. A000366, A218826. First column of A239895.

m = 20(*terms*); matc = Array[0&, {m, m}];
a366[n_] := (-2^(-1))^(n - 2)*Sum[Binomial[n, k]*(1 - 2^(n + k + 1))* BernoulliB[n + k + 1], {k, 0, n}];
ci[n_, k_] := ci[n, k] = Module[{v}, If[matc[[n, k]] == 0, If[n == k, v = 1, If[k == 1, v = c[n], v = Sum[Binomial[n - 1, i - 1]*c[i]*ci[n - i, k - 1], {i, 1, n - k + 1}]]]; matc[[n, k]] = v]; Return[matc[[n, k]]]];
a[n_] := a366[n + 1] - If[n == 1, 0, Sum[ci[n, i], {i, 2, n}]];
Array[a, m] (* Jean-François Alcover, Aug 03 2019, from PARI *)

a000366(n)= {return((-1/2)^(n-2)*sum(k=0, n, binomial(n, k)*(1-2^(n+k+1))*bernfrac(n+k+1)));}
ci(n,k) = {if (matc[n,k] == 0, if (n==k, v = 1, if (k==1, v = c(n), v = sum(i=1, n-k+1, binomial(n-1,i-1)*c(i)*ci(n-i,k-1)););); matc[n,k] = v;); return(matc[n,k]);}
c(n) = {if (n==1, return(a000366(n+1)), return(a000366(n+1) - sum(i=2, n, ci(n,i))));}
allc(m) = {matc = matrix(m,m); for (i=1, m, print1(c(i), ", "););}

A230740 O.g.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k(k+1)/2 + x) / (1 + k(k+1)/2*x).

Original entry on oeis.org

1, 1, 3, 10, 51, 370, 3691, 48525, 812089, 16832928, 422860609, 12649706416, 444120983433, 18078156682309, 844323149201499, 44838127594166770, 2686250544297734323, 180295858504407010026, 13473490672899749784979, 1114874245392058455432873

Offset: 0

Paul D. Hanna, Oct 28 2013

nonn

Compare to an o.g.f. of Genocchi numbers of the second kind (A000366):

Sum_{n>=0} x^n * Product_{k=1..n} k*(k+1)/2 / (1 + k*(k+1)/2*x).

			G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 51*x^4 + 370*x^5 + 3691*x^6 +...
where
A(x) = 1 + x*(1+x)/(1+x) + x^2*(1+x)*(3+x)/((1+x)*(1+3*x)) + x^3*(1+x)*(3+x)*(6+x)/((1+x)*(1+3*x)*(1+6*x)) + x^4*(1+x)*(3+x)*(6+x)*(10+x)/((1+x)*(1+3*x)*(1+6*x)*(1+10*x)) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..260

Cf. A230682.

{a(n)=polcoeff(sum(m=0, n, x^m*prod(k=1, m, k*(k+1)/2+x+x*O(x^n))/prod(k=1, m, 1+k*(k+1)/2*x+x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

a(n) ~ 2^(n+6) * n^(2*n+7/2) / (exp(2*n) * Pi^(2*n+5/2)). - Vaclav Kotesovec, Oct 28 2014

cp's OEIS Frontend

A130168 a(n) = (b(n) + b(n+1))/3, where b(n) = A000366(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula

A005439 Genocchi medians (or Genocchi numbers of second kind).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

Python

Sage

Sage

Formula

Extensions

A211183 Triangle T(n,k), 0<=k<=n, read by rows, given by (0, 1, 1, 3, 3, 6, 6, 10, 10, 15, ...) DELTA (1, 0, 2, 0, 3, 0, 4, 0, 5, ...) where DELTA is the operator defined in A084938.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A014784 Triangle of numbers associated with Genocchi numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A218826 Number of indecomposable (by concatenation) alternating n-anagrams.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A239894 Triangle read by rows: T(n,k) (n>=1, 1 <= k <= n) = number of alternating anagrams on n letters (of length 2n) which are decomposable into at most k slices.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A098278 D(n,0)/2^n, where D(n,x) is triangle A098277.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A230740 O.g.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k(k+1)/2 + x) / (1 + k(k+1)/2*x).