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A000111
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Euler or up/down numbers: e.g.f. sec(x) + tan(x). Also for n >= 2, half the number of alternating permutations on n letters (A001250).
(Formerly M1492 N0587)
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128
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1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521, 353792, 2702765, 22368256, 199360981, 1903757312, 19391512145, 209865342976, 2404879675441, 29088885112832, 370371188237525, 4951498053124096, 69348874393137901, 1015423886506852352, 15514534163557086905, 246921480190207983616, 4087072509293123892361
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refs;
listen;
history;
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internal format)
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OFFSET
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0,4
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COMMENTS
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Number of linear extensions of the "zig-zag" poset. See ch. 3, prob. 23 of Stanley. - Mitch Harris, Dec 27 2005
Number of increasing 0-1-2 trees on n vertices. [David Callan, Dec 22 2006]
Also the number of refinements of partitions. [Heinz-Richard Halder (halder.bichl(AT)t-online.de), Mar 07 2008]
The ratio a(n)/n! is also the probability that n numbers x1,x2,..,xn randomly chosen uniformly and independently in [0,1] satisfy x1>x2<x3>x4<...xn. [Pietro Majer, Jul 13 2009]
For n >= 2, a(n-2) = number of permutations w of an ordered n-set {x_1 < ... x_n} with the following properties: w(1)=x_n, w(n)=x_{n-1}, w(2)>w(n-1), and neither any subword of w, nor its reversal, has the first three properties. The count is unchanged if the third condition is replaced with w(2)<w(n-1). [Jeremy Martin (jmartin(AT)math.ku.edu), Mar 26 2010]
A partition of zigzag permutations of order n+1 by the smallest or the largest, whichever is behind. This partition has the same recurrent relation as increasing 1-2 trees of order n, by induction the bijection follows. - Wenjin Woan, May 06 2011
As can be seen from the asymptotics given in the FORMULA section, one has 2*n*a(n-1)/a(n) -> Pi as n->oo; see A132049/A132050 for the simplified fractions. - M. F. Hasler, Apr 03 2013
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REFERENCES
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D. André, Sur les permutations alternées, Journal de Mathématiques Pures et Appliquées, 7 (1881), 167-184.
Arnold, V. I., Bernoulli-Euler updown numbers associated with function singularities, their combinatorics and arithmetics, Duke Math. J. 63 (1991), 537-555.
M. D. Atkinson: Zigzag permutations and comparisons of adjacent elements, Information Processing Letters 21 (1985), 187-189.
M. D. Atkinson: Partial orders and comparison problems, Sixteenth Southeastern Conference on Combinatorics, Graph Theory and Computing, (Boca Raton, Feb 1985), Congressus Numerantium 47, 77-88.
P. Barry, A Note on Three Families of Orthogonal Polynomials defined by Circular Functions, and Their Moment Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.7.2. - From N. J. A. Sloane, Dec 27 2012
B. Bauslaugh and F. Ruskey, Generating alternating permutations lexicographically, Nordisk Tidskr. Informationsbehandling (BIT) 30 16-26 1990.
Brightwell, Graham; Cohen, Gerard; Fachini, Emanuela; Fairthorne, Marianne; Korner, Janos; Simonyi, Gabor; and Toth, Agnes Permutation capacities of families of oriented infinite paths. SIAM J. Discrete Math. 24 (2010), no. 2, 441-456.
L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 258-260, section #11.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 262.
C. Davis, Problem 4755, Amer. Math. Monthly, 64 (1957) 596; solution by W. J. Blundon, 65 (1958), 533-534. [Denoted by P_n in solution.]
Filippo Disanto and Thomas Wiehe, Some combinatorial problems on binary rooted trees occurring in population genetics, Arxiv preprint arXiv:1112.1295, 2011
H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 1965, p. 66.
Donaghey, Robert. Alternating permutations and binary increasing trees. J. Combinatorial Theory Ser. A 18 (1975), 141--148.MR0360299 (50 #12749) - From N. J. A. Sloane, Mar 19 2012
N. D. Elkies, On the sums Sum_{k = -infinity .. infinity} (4k+1)^(-n), Amer. Math. Monthly, 110 (No. 7, 2003), 561-573.
D. Foata and M.-P. Schutzenberger, Nombres d'Euler et permutations alternantes, in J. N. Srivastava et al., eds., A Survey of Combinatorial Theory (North Holland Publishing Company, Amsterdam, 1973), pp. 173-187.
Heinz-Richard Halder, Ueber Verfeinerungen von Partitionen, Periodica Mathematica Hungarica Vol. 12 (3), (1981), pp. 217-220.
O. Heimo and A. Karttunen, Series help-mates in 8, 9 and 10 moves (Problems 2901, 2974-2976), Suomen Tehtavaniekat (Proceedings of the Finnish Chess Problem Society) vol. 60, no. 2/2006, pp. 75, 77.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011; http://repository.wit.ie/1693/1/AoifeThesis.pdf
L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 238.
G. Kreweras, Les preordres totaux compatibles avec un ordre partiel. Math. Sci. Humaines No. 53 (1976), 5-30.
A. Mendes, A note on alternating permutations, Amer. Math. Monthly, 114 (2007), 437-440.
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 444.
F. Murtagh, "Counting dendrograms: a survey", Discrete Applied Mathematics, 7 (1984), 191-199.
E. Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, p. 110.
E. Norton, Symplectic Reflection Algebras in Positive Characteristic as Ore Extensions, arXiv preprint arXiv:1302.5411, 2013
C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 184.
Remmel, Jeffrey B. Generating functions for alternating descents and alternating major index. Ann. Comb. 16 (2012), no. 3, 625--650. MR2960023. - From N. J. A. Sloane, Jan 15 2013
Y. Sano, The principal numbers of K. Saito for the types A_l, D_l and E_l, Discr. Math., 307 (2007), 2636-2642.
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, vol. 7 (1877), 157-187.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. Staib, Trigonometric power series, Math. Mag., 49 (1976), 147-148.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1997 and Vol. 2, 1999; see Problem 5.7.
Zhi-Hong Sun, Congruences involving Bernoulli polynomials, Discr. Math., 308 (2007), 71-112.
A. Vieru, Agoh's conjecture: its proof, its generalizations, its analogues, Arxiv preprint arXiv:1107.2938, 2011.
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LINKS
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N. J. A. Sloane, The first 200 Euler numbers: Table of n, a(n) for n = 0..199
Joerg Arndt, Fxtbook, pp. 281-282
J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles.
F. Bergeron, M. Bousquet-Mélou and S. Dulucq, Standard paths in the composition poset
David Callan, A note on downup permutations and increasing 0-1-2 trees
Filippo Disanto, Andre' permutations of the second kind associated to strictly binary increasing trees and left to right minima in their sub-permutations, Arxiv preprint arXiv:1202.1139, 2012
N. D. Elkies, On the sums Sum((4k+1)^(-n),k,-inf,+inf)
N. D. Elkies, New Directions in Enumerative Chess Problems, The Electronic Journal of Combinatorics, vol. 11(2), 2004.
P. Flajolet, S. Gerhold and B. Salvy, On the non-holonomic character of logarithms, powers and the n-th prime function
S. N. Gladkovskii, On the continued fraction expansion for functions 1/sin(x) + cot(x) and sec(x) + tan(x), arXiv:1208.2243v1 [math.HO]
M. Josiat-Verges, Enumeration of snakes and cycle-alternating permutations
Dmitry Kruchinin, Integer properties of a composition of exponential generating functions, arXiv:1211.2100
Kruchinin, Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565
Peter Luschny, An old operation on sequences: the Seidel transform
J. L. Martin and J. D. Wagner, Updown numbers and the initial monomials of the slope variety, Electronic J. Combin. 16, no. 1 (2009), Research Paper R82. [From Jeremy Martin (jmartin(AT)math.ku.edu), Mar 26 2010]
J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A 44-54 1996 (Abstract, pdf, ps).
A. Randrianarivony and J. Zeng, Sur une extension des nombres d'Euler et les records des permutations alternantes, J. Combin. Theory Ser. A 68 (1994), 68-99.
A. Randrianarivony and J. Zeng, Une famille des polynomes qui interpole plusieurs suites..., Adv. Appl. Math. 17 (1996), 1-26.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
R. P. Stanley, Queue problems revisited, Suomen Tehtavaniekat (Proceedings of the Finnish Chess Problem Society), vol. 59, no. 4 (2005), 193-203.
R. P. Stanley, Permutations
Ross Tang, An Explicit Formula for the Euler zigzag numbers (Up/down numbers) from power series [From Ross Tang (ph.tchaa(AT)gmail.com), Jul 28 2010]
Eric Weisstein's World of Mathematics, Euler Zigzag Number
Eric Weisstein's World of Mathematics, Alternating Permutation
Eric Weisstein's World of Mathematics, Entringer Number
Index entries for "core" sequences
Index entries for sequences related to boustrophedon transform
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FORMULA
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E.g.f.: (1+sin(x))/cos(x) = tan(x) + sec(x).
E.g.f. for a(n+1) is 1/(cos(x/2)-sin(x/2))^2 = 1/(1-sin(x)) = d/dx(sec(x)+tan(x)).
E.g.f. A(x)=-log(1-sin(x), for a(n+1) [From Kruchinin Vladimir, Aug 09 2010]
O.g.f.: A(x) = 1+x/(1-x-x^2/(1-2*x-3*x^2/(1-3*x-6*x^2/(1-4*x-10*x^2/(1-... -n*x-(n*(n+1)/2)*x^2/(1- ...)))))) (continued fraction). - Paul D. Hanna, Jan 17 2006
O.g.f. A(x) = y satisfies 2y' = 1 + y^2. - Michael Somos, Feb 03 2004
a(n) = P_n(0) + Q_n(0) (see A155100 and A104035), defining Q_{-1} = 0. Cf. A156142.
2*a(n+1) = Sum_{k=0..n} binomial(n, k)*a(k)*a(n-k).
Asymptotics: a(n) ~ 2^(n+2)*n!/Pi^(n+1).
a(n) = (n-1)*a(n-1) - sum{i=2, n-2, (i-1)*E(n-1, i)}, where E are the Entringer numbers A008280. - Jon Perry, Jun 09 2003
a(2*k) = (-1)^k euler(2k) and a(2k-1) = (-1)^(k-1)2^(2k)(2^(2k)-1) bernoulli(2k)/(2k) - C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 17 2005
|a(n+1)-2*a(n)|=A000708(n) . - Philippe DELEHAM, Jan 13 2007
a(n) = 2^n|E(n,1/2)+E(n,1)| where E(n,x) are the Euler polynomials. [Peter Luschny, Jan 25 2009]
a(n) = 2^{n+2}*n!*S(n+1)/(Pi)^{n+1}, where S(n)=Sum(1/(4k+1)^n, k=-inf..inf) (see the Elkies reference). [From Emeric Deutsch, Aug 17 2009]
a(n) = i^{n+1} sum_{k=1..n+1} sum_{j=0..k} binomial(k,j)(-1)^j (k-2j)^{n+1} (2i)^{-k} k^{-1} [From Ross Tang (ph.tchaa(AT)gmail.com), Jul 28 2010]
a(n)=sum((if evenp(n+k) then (-1)^((n+k)/2)*sum(j!*stirling2(n,j)*2^(1-j)*(-1)^(n+j-k)*binomial(j-1,k-1),j,k,n) else 0),k,1,n), n>0. [From Vladimir Kruchinin, Aug 19 2010]
If n==1(mod 4) is prime, then a(n)==1(mod n); if n==3(mod 4) is prime, then a(n)==-1(mod n). [From Vladimir Shevelev, Aug 31 2010]
For m>=0, a(2^m)==1(mod 2^m); If p is prime, then a(2*p)==1(mod 2*p). [From Vladimir Shevelev, Sep 03 2010]
From Peter Bala, Jan 26 2011: (Start)
a(n) = A(n,I)/(1+I)^(n-1), where I = sqrt(-1) and {A(n,x)}n>=1 = [1,1+x,1+4*x+x^2,1+11*x+11*x^2+x^3,...] denotes the sequence of Eulerian polynomials.
Equivalently, a(n) = I^(n+1)*sum {k=1..n} (-1)^k*k!*Stirling2(n,k) * ((1+I)/2)^(k-1) = I^(n+1)*sum {k = 1..n} (-1)^k*((1+I)/2)^(k-1)* sum{j = 0..k} (-1)^(k-j)*binomial(k,j)*j^n.
This explicit formula for a(n) can be used to obtain congruence results. For example, for odd prime p, a(p) = (-1)^((p-1)/2) (mod p), as noted by Vladimir Shevelev above.
For the corresponding type B results see A001586. For the corresponding results for plane increasing 0-1-2 trees see A080635.
For generalized Eulerian, Stirling and Bernoulli numbers associated with the zigzag numbers see A145876, A147315 and A185424, respectively. For a recursive triangle to calculate a(n) see A185414.
(End)
a(n) = I^(n+1)*2*Li_{-n}(-I) for n > 0. Li_{s}(z) is the polylogarithm. [Peter Luschny, Jul 29 2011]
a(n)=2*sum(m=0..(n-2)/2, 4^m*(sum(i=m..(n-1)/2, (i-(n-1)/2)^(n-1)*binomial(n-2*m-1,i-m)*(-1)^(n-i-1)))), n>1, a(0)=1, a(1)=1. [From Vladimir Kruchinin, Aug 09 2011]
E.g.f.: tan(x)+sec(x)=1+x/U(0); U(k)= 4k+1-x/(2-x/(4k+3+x/(2+x/U(k+1))));(continued fraction). - Sergei N. Gladkovskii, Nov 14 2011
a(n) = D^(n-1)(1/(1-x)) evaluated at x = 0, where D is the operator sqrt(1-x^2)*d/dx. Cf. A006154. a(n) equals the alternating sum of the nonzero elements of row n-1 of A196776. This leads to a combinatorial interpretation for a(n); for example, a(4*n+2) counts the number of ordered set partitions of 4*n+1 into k odd-sized blocks, k = 1 (mod 4), minus the number of ordered set partitions of 4*n+1 into k odd-sized blocks, k = 3 (mod 4). Cf A002017 - Peter Bala, Dec 06 2011
E.g.f. for a(n+1) is E(x)=1/(1-sin(x))=1 + x/(1 - x + x^2/G(0)) ; G(k)= (2*k+2)*(2*k+3)-x^2+(2*k+2)*(2*k+3)*x^2/G(k+1); (continued fraction Euler's kind, 1-step ). - Sergei N. Gladkovskii, Jan 06 2012
E.g.f. for a(n+1) is E(x)=1/(1-sin(x))=1/(1 - x/(1 + x^2/G(0)) ; G(k)= 8*k+6-x^2/(1 + (2*k+2)*(2*k+3)/G(k+1)); (continued fraction Euler's 2 kind, 2-step ). - Sergei N. Gladkovskii, Jan 06 2012
E.g.f. for a(n+1) is E(x)= 1/(1 - sin(x))=1/(1 - x*G(0)) ; G(k)= 1 - x^2/(2*(2*k+1)*(4*k+3) - 2*x^2*(2*k+1)*(4*k+3)/(x^2 - 4*(k+1)*(4*k+5)/G(k+1))); (continued fraction 3 kind, 3-step ). - Sergei N. Gladkovskii, Jan 06 2012
E.g.f. for a(n+1) is E(x)= 1/(1 - sin(x))=1/(1 - x*G(0)) where G(k)= 1 + x^2/( (2*k+1)*(2*k+3) - 2*k+1)*(2*k+3)^2/(2*k+3 - (2*k+2)/G(k+1))) ; (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Oct 01 2012
E.g.f.: tan(x)+sec(x)=1+2*x/(U(0)-x) where U(k)= 4k+2 - x^2/U(k+1);(continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 27 2012
E.g.f.: tan(x)+sec(x) = 1+2*x/(2*U(0)-x) where U(k) = 4*k+1 - x^2/(16*k+12 - x^2/U(k+1)); (continued fraction). - Sergei N. Gladkovskii, Nov 11 2012
E.g.f.: tan(x)+ sec(x) = 4/(2-x*G(0))-1 where G(k)=1 - x^2/(x^2 - 4*(2*k+1)*(2*k+3)/G(k+1) ); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 10 2012
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EXAMPLE
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1 + x + x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 61*x^6 + 272*x^7 + 1385*x^8 + ...
Sequence starts 1,1,2,5,16,... since possibilities are {}, {A}, {AB}, {ACB, BCA}, {ACBD, ADBC, BCAD, BDAC, CDAB}, {ACBED, ADBEC, ADCEB, AEBDC, AECDB, BCAED, BDAEC, BDCEA, BEADC, BECDA, CDAEB, CDBEA, CEADB, CEBDA, DEACB, DEBCA}, etc. - Henry Bottomley, Jan 17 2001
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MAPLE
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A000111 := n-> n!*coeff(series(sec(x)+tan(x), x, n+1), x, n);
s := series(sec(x)+tan(x), x, 100): A000111 := n-> n!*coeff(s, x, n);
A000111:=n->piecewise(n mod 2=1, (-1)^((n-1)/2)*2^(n+1)*(2^(n+1)-1)*bernoulli(n+1)/(n+1), (-1)^(n/2)*euler(n)):seq(A000111(n), n=0..30); A000111:=proc(n) local k: k:=floor((n+1)/2): if n mod 2=1 then RETURN((-1)^(k-1)*2^(2*k)*(2^(2*k)-1)*bernoulli(2*k)/(2*k)) else RETURN((-1)^k*euler(2*k)) fi: end:seq(A000111(n), n=0..30); (C. Ronaldo)
T := n -> 2^n*abs(euler(n, 1/2)+euler(n, 1)): [Peter Luschny, Jan 25 2009]
S := proc(n, k) option remember; if k=0 then RETURN(`if`(n=0, 1, 0)) fi; S(n, k-1)+S(n-1, n-k) end:
A000364 := n -> S(2*n, 2*n);
A000182 := n -> S(2*n+1, 2*n+1);
A000111 := n -> S(n, n); [Peter Luschny, Jul 29 2009]
a := proc (n) options operator, arrow: 2^(n+2)*factorial(n)*(sum(1/(4*k+1)^(n+1), k = -infinity .. infinity))/Pi^(n+1) end proc: 1, seq(a(n), n = 1 .. 22); [From Emeric Deutsch, Aug 17 2009]
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MATHEMATICA
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n=22; CoefficientList[Series[(1+Sin[x])/Cos[x], {x, 0, n}], x] * Table[k!, {k, 0, n}] (* From Jean-François Alcover, May 18 2011, after M. Somos *)
a[n_] := If[EvenQ[n], Abs[EulerE[n]], Abs[(2^(n+1)*(2^(n+1)-1)*BernoulliB[n+1])/(n+1)]]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Oct 09 2012, after C. Ronaldo *)
ee = Table[ 2^n*EulerE[n, 1] + EulerE[n] - 1, {n, 0, 26}]; Table[ Differences[ee, n] // First // Abs, {n, 0, 26}] (* Jean-François Alcover, Mar 21 2013, after Paul Curtz *)
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PROG
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# From Michael Somos, Feb 03 2004: (Start)
(PARI) {a(n) = if( n<1, n==0, n--; n! * polcoeff( 1 / (1 - sin(x + x * O(x^n))), n))}
(PARI) {a(n) = local(v=[1], t); if( n<0, 0, for(k=2, n+2, t=0; v = vector(k, i, if( i>1, t+= v[k+1-i]))); v[2])}
(PARI) {a(n) = local(an); if( n<1, n>=0, an = vector(n+1, m, 1); for( m=2, n, an[m+1] = sum( k=0, m-1, binomial(m-1, k) * an[k+1] * an[m-k]) / 2); an[n+1])}
# (End)
(PARI) z='z+O('z^66); egf = (1+sin(z))/cos(z); Vec(serlaplace(egf)) /* Joerg Arndt, Apr 30 2011 */
(PARI) A000111(n)={my(k); sum(m=0, n\2, (-1)^m*sum(j=0, k=n+1-2*m, binomial(k, j)*(-1)^j*(k-2*j)^(n+1))/k>>k)} \\ M. F. Hasler, May 19 2012
(PARI) A000111(n)=if(n, 2*abs(polylog(-n, I)), 1) \\ M. F. Hasler, May 20 2012
(Maxima) a(n):=sum((if evenp(n+k) then (-1)^((n+k)/2)*sum(j!*stirling2(n, j)*2^(1-j)*(-1)^(n+j-k)*binomial(j-1, k-1), j, k, n) else 0), k, 1, n); [From Kruchinin Vladimir, Aug 19 2010]
(Maxima)
a(n):=if n<2 then 1 else 2*sum(4^m*(sum((i-(n-1)/2)^(n-1)*binomial(n-2*m-1, i-m)*(-1)^(n-i-1), i, m, (n-1)/2)), m, 0, (n-2)/2); [From Vladimir Kruchinin, Aug 09 2011]
(Sage) # Algorithm of L. Seidel (1877)
def A000111_list(n) :
R = []; A = {-1:0, 0:1}; k = 0; e = 1
for i in (0..n) :
Am = 0; A[k + e] = 0; e = -e
for j in (0..i) : Am += A[k]; A[k] = Am; k += e
R.append(A[-i//2] if i%2 == 0 else A[i//2])
return R
A000111_list(22) # Peter Luschny, March 31 2012
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CROSSREFS
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Cf. A000364 (secant numbers), A000182 (tangent numbers). See also A008280, A008281, A008282, A010094, A059720 for related triangles.
A diagonal of A008970.
Cf. A109449 for an extension to an exponential Riordan array.
Sequence in context: A178123 A138265 A163747 * A007976 A058259 A033543
Adjacent sequences: A000108 A000109 A000110 * A000112 A000113 A000114
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KEYWORD
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nonn,core,eigen,nice,easy,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Edited by M. F. Hasler, Apr 04 2013
Corrected title - Geoffrey Critzer, May 18 2013
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STATUS
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approved
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