A059419 -id:A059419 - cp's OEIS frontend

A000182 Tangent (or "Zag") numbers: e.g.f. tan(x), also (up to signs) e.g.f. tanh(x).

1, 2, 16, 272, 7936, 353792, 22368256, 1903757312, 209865342976, 29088885112832, 4951498053124096, 1015423886506852352, 246921480190207983616, 70251601603943959887872, 23119184187809597841473536, 8713962757125169296170811392, 3729407703720529571097509625856

Offset: 1

N. J. A. Sloane

Number of Joyce trees with 2n-1 nodes. Number of tremolo permutations of {0,1,...,2n}. - Ralf Stephan, Mar 28 2003
The Hankel transform of this sequence is A000178(n) for n odd = 1, 12, 34560, ...; example: det([1, 2, 16; 2, 16, 272, 16, 272, 7936]) = 34560. - Philippe Deléham, Mar 07 2004
a(n) is the number of increasing labeled full binary trees with 2n-1 vertices. Full binary means every non-leaf vertex has two children, distinguished as left and right; labeled means the vertices are labeled 1,2,...,2n-1; increasing means every child has a label greater than its parent. - David Callan, Nov 29 2007
From Micha Hofri (hofri(AT)wpi.edu), May 27 2009: (Start)
a(n) was found to be the number of permutations of [2n] which when inserted in order, to form a binary search tree, yield the maximally full possible tree (with only one single-child node).
The e.g.f. is sec^2(x)=1+tan^2(x), and the same coefficients can be manufactured from the tan(x) itself, which is the e.g.f. for the number of trees as above for odd number of nodes. (End)
a(n) is the number of increasing strict binary trees with 2n-1 nodes. For more information about increasing strict binary trees with an associated permutation, see A245894. - Manda Riehl, Aug 07 2014
For relations to alternating permutations, Euler and Bernoulli polynomials, zigzag numbers, trigonometric functions, Fourier transform of a square wave, quantum algebras, and integrals over and in n-dimensional hypercubes and over Green functions, see Hodges and Sukumar. For further discussion on the quantum algebra, see the later Hodges and Sukumar reference and the paper by Hetyei presenting connections to the general combinatorial theory of Viennot on orthogonal polynomials, inverse polynomials, tridiagonal matrices, and lattice paths (thereby related to continued fractions and cumulants). - Tom Copeland, Nov 30 2014
The Zigzag Hankel transform is A000178. That is, A000178(2*n - k) = det( [a(i+j - k)]{i,j = 1..n} ) for n>0 and k=0,1. - _Michael Somos, Mar 12 2015
a(n) is the number of standard Young tableaux of skew shape (n,n,n-1,n-2,...,3,2)/(n-1,n-2,n-3,...,2,1). - Ran Pan, Apr 10 2015
For relations to the Sheffer Appell operator calculus and a Riccati differential equation for generating the Meixner-Pollaczek and Krawtchouk orthogonal polynomials, see page 45 of the Feinsilver link and Rzadkowski. - Tom Copeland, Sep 28 2015
For relations to an elliptic curve, a Weierstrass elliptic function, the Lorentz formal group law, a Lie infinitesimal generator, and the Eulerian numbers A008292, see A155585. - Tom Copeland, Sep 30 2015
Absolute values of the alternating sums of the odd-numbered rows (where the single 1 at the apex of the triangle is counted as row #1) of the Eulerian triangle, A008292. The actual alternating sums alternate in sign, e.g., 1, -2, 16, -272, etc. (Even-numbered rows have alternating sums always 0.) - Gregory Gerard Wojnar, Sep 28 2018
The sequence is periodic modulo any odd prime p. The minimal period is (p-1)/2 if p == 1 mod 4 and p-1 if p == 3 mod 4 [Knuth & Buckholtz, 1967, Theorem 1]. - Allen Stenger, Aug 03 2020
From Peter Bala, Dec 24 2021: (Start)
Conjectures:
1) The sequence taken modulo any integer k eventually becomes periodic with period dividing phi(k).
2) The Gauss congruences a(n*p^k) == a(n*p^(k-1)) ( mod p^k ) hold for all prime p and positive integers n and k, except when p = 2, n = 1 and k = 1 or 2.
3) For i >= 1 define a_i(n) = a(n+i). The Gauss congruences a_i(n*p^k) == a_i(n*p^(k-1)) ( mod p^k ) hold for all prime p and positive integers n and k. If true, then for each i >= 1 the expansion of exp(Sum_{n >= 1} a_i(n)*x^n/n) has integer coefficients. For an example, see A262145.(End)

			tan(x) = x + 2*x^3/3! + 16*x^5/5! + 272*x^7/7! + ... = x + 1/3*x^3 + 2/15*x^5 + 17/315*x^7 + 62/2835*x^9 + O(x^11).
tanh(x) = x - 1/3*x^3 + 2/15*x^5 - 17/315*x^7 + 62/2835*x^9 - 1382/155925*x^11 + ...
(sec x)^2 = 1 + x^2 + 2/3*x^4 + 17/45*x^6 + ...
a(3)=16 because we have: {1, 3, 2, 5, 4}, {1, 4, 2, 5, 3}, {1, 4, 3, 5, 2},
  {1, 5, 2, 4, 3}, {1, 5, 3, 4, 2}, {2, 3, 1, 5, 4}, {2, 4, 1, 5, 3},
  {2, 4, 3, 5, 1}, {2, 5, 1, 4, 3}, {2, 5, 3, 4, 1}, {3, 4, 1, 5, 2},
  {3, 4, 2, 5, 1}, {3, 5, 1, 4, 2}, {3, 5, 2, 4, 1}, {4, 5, 1, 3, 2},
  {4, 5, 2, 3, 1}. - _Geoffrey Critzer_, May 19 2013

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 932.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 111.
H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 1965, p. 69.
L. M. Milne-Thompson, Calculus of Finite Differences, 1951, p. 148 (the numbers |C^{2n-1}|).
J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, p. 282.
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 444.
H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1, p. 20.
L. Seidel, Über eine einfache Entstehungsweise der Bernoullischen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 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).
E. van Fossen Conrad, Some continued fraction expansions of elliptic functions, PhD thesis, The Ohio State University, 2002, p. 28.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, pp. 267-268.

Seiichi Manyama, Table of n, a(n) for n = 1..243 (terms 1..100 from N. J. A. Sloane)
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Étienne Bellin, Arthur Blanc-Renaudie, Emmanuel Kammerer, and Igor Kortchemski, Uniform attachment with freezing, arXiv:2308.00493 [math.PR], 2023. See p. 13.
Beáta Bényi, Miguel Méndez, José L. Ramírez, and Tanay Wakhare, Restricted r-Stirling Numbers and their Combinatorial Applications, arXiv:1811.12897 [math.CO], 2018.
Richard P. Brent and David Harvey, Fast computation of Bernoulli, Tangent and Secant numbers, arXiv preprint arXiv:1108.0286 [math.CO], 2011.
F. C. S. Brown, T. M. A. Fink and K. Willbrand, On arithmetic and asymptotic properties of up-down numbers, arXiv:math/0607763 [math.CO], 2006.
K.-W. Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.
Suyuong Choi and Younghan Yoon, A decomposition of graph a-numbers, arXiv:2508.06855 [math.CO], 2025. See p. 13.
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 p. 11.
D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.
D. Dumont and G. Viennot, A combinatorial interpretation of the Seidel generation of Genocchi numbers, Preprint, Annotated scanned copy.
A. L. Edmonds and S. Klee, The combinatorics of hyperbolized manifolds, arXiv preprint arXiv:1210.7396 [math.CO], 2012. - From _N. J. A. Sloane_, Jan 02 2013
P. Feinsilver, Lie algebras, representations, and analytic semigroups through dual vector fields [broken link]
C. J. Fewster and D. Siemssen, Enumerating Permutations by their Run Structure, arXiv preprint arXiv:1403.1723 [math.CO], 2014.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 144.
Dominique Foata and Guo-Niu Han, Doubloons and new q-tangent numbers, Quart. J. Math. 62 (2) (2011) 417-432.
Dominique Foata and Guo-Niu Han, Tree Calculus for Bivariable Difference Equations, Journal of Difference Equations and Applications, 20 (2014), 1453-1488; arXiv:1304.2484 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 02 2013
Dominique Foata and Guo-Niu Han, Seidel Triangle Sequences and Bi-Entringer Numbers, European Journal of Combinatorics, 42 (2014), 243-260.
Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
M.-P. Grosset and A. P. Veselov, Bernoulli numbers and solitons, arXiv:math/0503175 [math.GM], 2005.
Christian Günther and Kai-Uwe Schmidt, L^q norms of Fekete and related polynomials, arXiv:1602.01750 [math.NT], 2016.
Guo-Niu Han and Jing-Yi Liu, Combinatorial proofs of some properties of tangent and Genocchi numbers, European Journal of Combinatorics, Vol. 71 (2018), pp. 99-110; arXiv preprint, arXiv:1707.08882 [math.CO], 2017-2018.
Yanjun Han and Jonathan Niles-Weed, Approximate independence of permutation mixtures, arXiv:2408.09341 [math.ST], 2024. See p. 12.
G. Hetyei, Meixner polynomials of the second kind and quantum algebras representing su(1,1), arXiv preprint arXiv:0909.4352 [math.QA], 2009.
A. Hodges and C. Sukumar, Bernoulli, Euler, permutations and quantum algebras, Proc. Royal Soc. A (2007) 463, 2401-2414.
A. Hodges and C. Sukumar, Quantum algebras and parity-dependent spectra, Proc. Royal Soc. A (2007) 463, 2415-2427.
Jan Hubička, Matěj Konečný, Štěpán Vodseďálek, and Andy Zucker, Counting big Ramsey degrees of the homogeneous and universal K_4-free graph, arXiv:2505.22620 [math.CO], 2025. See pp. 2, 4.
Hsien-Kuei Hwang and Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.
Svante Janson, Euler-Frobenius numbers and rounding, arXiv preprint arXiv:1305.3512 [math.PR], 2013
Donald E. Knuth and Thomas J. Buckholtz, Computation of tangent, Euler and Bernoulli numbers, Math. Comp. 21 1967 663-688.
D. E. Knuth and Thomas J. Buckholtz, Computation of tangent, Euler and Bernoulli numbers, Math. Comp. 21 1967 663-688. [Annotated scanned copy]
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Johann Heinrich Lambert, Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques, Histoire de l'Académie Royale des Sciences et des Belles-Lettres de Berlin 1761 (Berlin: Haude et Spener, 1768) pp. 265-322. [broken link]
F. Luca and P. Stanica, On some conjectures on the monotonicity of some arithmetical sequences, J. Combin. Number Theory 4 (2012) 1-10.
Peter Luschny, Approximation, inclusion and asymptotics of the Euler numbers
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N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924 [Annotated scanned copy of pages 144-151 and 456-463]
Jay Rosen, The Number of Product-Weighted Lead Codes for Ballots and Its Relation to the Ursell Functions of the Linear Ising Model, Journal of Combinatorial Theory, Vol. 20, No.3, May 1976, 377-384.
G. Rzadkowski, Bernoulli numbers and solitons-revisited, Jrn. Nonlinear Math. Physics, 1711, pp. 121-126. (Added by Tom Copeland, Sep 29 2015)
Raphael Schumacher, Rapidly Convergent Summation Formulas involving Stirling Series, arXiv preprint arXiv:1602.00336 [math.NT], 2016.
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N. J. A. Sloane, Rough notes on Genocchi numbers
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R. P. Stanley, Permutations
Ross Street, Trees, permutations and the tangent function gives definition of Joyce trees and tremolo permutations, arXiv:math/0303267 [math.HO], 2003.
Ross Street, Surprising relationships connecting ploughing a field, mathematical trees, permutations, and trigonometry, Slides from a talk, July 15 2015, Macquarie University. ["There is a Web Page oeis.org by N. J. A. Sloane. It tells, from typing the first few terms of a sequence, whether that sequence has occurred somewhere else in Mathematics. Postgraduate student Daniel Steffen traced this down and found, to our surprise, that the sequence was related to the tangent function tan x. Ryan and Tam searched out what was known about this connection and discovered some apparently new results. We all found this a lot of fun and I hope you will too."]
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Index entries for "core" sequences
Index entries for sequences related to boustrophedon transform
Index entries for sequences related to Bernoulli numbers.

A350972 is essentially the same sequence.
a(n)=2^(n-1)*A002105(n). Apart from signs, 2^(2n-2)*A001469(n) = n*a(n).
Cf. A001469, A002430, A036279, A000364 (secant numbers), A000111 (secant-tangent numbers), A024283, A009764. First diagonal of A059419 and of A064190.
Cf. A009006, A009725, A029584, A012509, A009123, A009567.
Equals A002425(n) * 2^A101921(n).
Equals leftmost column of A162005. - Johannes W. Meijer, Jun 27 2009
Cf. A258880, A258901. Cf. A002105, A080635, A234797.
Cf. A019538, A110501.

series(tan(x),x,40);
with(numtheory): a := n-> abs(2^(2*n)*(2^(2*n)-1)*bernoulli(2*n)/(2*n));
A000182_list := proc(n) local T,k,j; T[1] := 1;
for k from 2 to n do T[k] := (k-1)*T[k-1] od;
   for k from 2 to n do
       for j from k to n do
           T[j] := (j-k)*T[j-1]+(j-k+2)*T[j] od od;
seq(T[j], j=1..n)  end:
A000182_list(15);  # Peter Luschny, Apr 02 2012

Table[ Sum[2^(2*n + 1 - k)*(-1)^(n + k + 1)*k!*StirlingS2[2*n + 1, k], {k, 1, 2*n + 1}], {n, 0, 7}] (* Victor Adamchik, Oct 05 2005 *)
v[1] = 2; v[n_] /; n >= 2 := v[n] = Sum[ Binomial[2 n - 3, 2 k - 2] v[k] v[n - k], {k, n - 1}]; Table[ v[n]/2, {n, 15}] (* Zerinvary Lajos, Jul 08 2009 *)
Rest@ Union[ Range[0, 29]! CoefficientList[ Series[ Tan[x], {x, 0, 30}], x]] (* Harvey P. Dale, Oct 19 2011; modified by Robert G. Wilson v, Apr 02 2012 *)
t[1, 1] = 1; t[1, 0] = 0; t[n_ /; n > 1, m_] := t[n, m] = m*(m+1)*Sum[t[n-1, k], {k, m-1, n-1}]; a[n_] := t[n, 1]; Table[a[n], {n, 1, 15}]  (* Jean-François Alcover, Jan 02 2013, after A064190 *)
a[ n_] := If[ n < 1, 0, With[{m = 2 n - 1}, m! SeriesCoefficient[ Tan[x], {x, 0, m}]]]; (* Michael Somos, Mar 12 2015 *)
a[ n_] := If[ n < 1, 0, ((-16)^n - (-4)^n) Zeta[1 - 2 n]]; (* Michael Somos, Mar 12 2015 *)
Table[2 PolyGamma[2n - 1, 1/2]/Pi^(2n), {n, 1, 10}] (* Vladimir Reshetnikov, Oct 18 2015 *)
a[ n_] := a[n] = If[ n < 2, Boole[n == 1], Sum[Binomial[2 n - 2, 2 k - 1] a[k] a[n - k], {k, n - 1}]]; (* Michael Somos, Aug 02 2018 *)
a[n_] := (2^(2*n)*(2^(2*n) - 1)*Abs[BernoulliB[2*n]])/(2*n); a /@  Range[20]  (* Stan Wagon, Nov 21 2022 *)

a(n):=sum(sum(binomial(k,r)*sum(sum(binomial(l,j)/2^(j-1)*sum((-1)^(n)*binomial(j,i)*(j-2*i)^(2*n),i,0,floor((j-1)/2))*(-1)^(l-j),j,1,l)*(-1)^l*binomial(r+l-1,r-1),l,1,2*n)*(-1)^(1-r),r,1,k)/k,k,1,2*n); /* Vladimir Kruchinin, Aug 23 2010 */

a[n]:=if n=1 then 1 else 2*sum(sum(binomial(2*j,j+k)*(-4*k^2)^(n-1)*(-1)^k/(4^j),k,1,j),j,1,n-1);
makelist(a[n],n,1,30); /* Tani Akinari, Sep 20 2023 */

{a(n) = if( n<1, 0, ((-4)^n - (-16)^n) * bernfrac(2*n) / (2*n))};

{a(n) = my(an); if( n<2, n==1, an = vector(n, m, 1); for( m=2, n, an[m] = sum( k=1, m-1, binomial(2*m - 2, 2*k - 1) * an[k] * an[m-k])); an[n])}; /* Michael Somos */

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

{a(n) = my(X=x+x*O(x^n),Egf); Egf = x*sum(m=0,n, prod(k=1,m, tanh(2*k*X))); (n-1)!*polcoeff(Egf,n)} /* Paul D. Hanna, May 11 2010 */

/* Continued Fraction for the e.g.f. tan(x), from Paul D. Hanna: */
{a(n)=local(CF=1+O(x)); for(i=1, n, CF=1/(2*(n-i+1)-1-x^2*CF)); (2*n-1)!*polcoeff(x*CF, 2*n-1)}

/* O.g.f. Sum_{n>=1} a(n)*x^n, from Paul D. Hanna Feb 05 2013: */
{a(n)=polcoeff( x+2*x*sum(m=1, n, x^m*prod(k=1, m, (2*k-1)^2/(1+(2*k-1)^2*x +x*O(x^n))) ), n)}

# The objective of this implementation is efficiency.
# n -> [0, a(1), a(2), ..., a(n)] for n > 0.
def A000182_list(n):
    T = [0 for i in range(1, n+2)]
    T[1] = 1
    for k in range(2, n+1):
        T[k] = (k-1)*T[k-1]
    for k in range(2, n+1):
        for j in range(k, n+1):
            T[j] = (j-k)*T[j-1]+(j-k+2)*T[j]
    return T
print(A000182_list(100)) # Peter Luschny, Aug 07 2011

from sympy import bernoulli
def A000182(n): return abs(((2-(2<<(m:=n<<1)))*bernoulli(m)<Chai Wah Wu, Apr 14 2023

# Algorithm of L. Seidel (1877)
# n -> [a(1), ..., a(n)] for n >= 1.
def A000182_list(len) :
    R = []; A = {-1:0, 0:1}; k = 0; e = 1
    for i in (0..2*len-1) :
        Am = 0; A[k + e] = 0; e = -e
        for j in (0..i) : Am += A[k]; A[k] = Am; k += e
        if e > 0 : R.append(A[i//2])
    return R
A000182_list(15) # Peter Luschny, Mar 31 2012

E.g.f.: log(sec x) = Sum_{n > 0} a(n)*x^(2*n)/(2*n)!.
E.g.f.: tan x = Sum_{n >= 0} a(n+1)*x^(2*n+1)/(2*n+1)!.
E.g.f.: (sec x)^2 = Sum_{n >= 0} a(n+1)*x^(2*n)/(2*n)!.
2/(exp(2x)+1) = 1 + Sum_{n>=1} (-1)^(n+1) a(n) x^(2n-1)/(2n-1)! = 1 - x + x^3/3 - 2*x^5/15 + 17*x^7/315 - 62*x^9/2835 + ...
a(n) = 2^(2*n) (2^(2*n) - 1) |B_(2*n)| / (2*n) where B_n are the Bernoulli numbers (A000367/A002445 or A027641/A027642).
Asymptotics: a(n) ~ 2^(2*n+1)*(2*n-1)!/Pi^(2*n).
Sum[2^(2*n + 1 - k)*(-1)^(n + k + 1)*k!*StirlingS2[2*n + 1, k], {k, 1, 2*n + 1}]. - Victor Adamchik, Oct 05 2005
a(n) = abs[c(2*n-1)] where c(n)= 2^(n+1) * (1-2^(n+1)) * Ber(n+1)/(n+1) = 2^(n+1) * (1-2^(n+1)) * (-1)^n * Zeta(-n) = [ -(1+EN(.))]^n = 2^n * GN(n+1)/(n+1) = 2^n * EP(n,0) = (-1)^n * E(n,-1) = (-2)^n * n! * Lag[n,-P(.,-1)/2] umbrally = (-2)^n * n! * C{T[.,P(.,-1)/2] + n, n} umbrally for the signed Euler numbers EN(n), the Bernoulli numbers Ber(n), the Genocchi numbers GN(n), the Euler polynomials EP(n,t), the Eulerian polynomials E(n,t), the Touchard / Bell polynomials T(n,t), the binomial function C(x,y) = x!/[(x-y)!*y! ] and the polynomials P(j,t) of A131758. - Tom Copeland, Oct 05 2007
a(1) = A094665(0,0)*A156919(0,0) and a(n) = Sum_{k=1..n-1} 2^(n-k-1)*A094665(n-1, k)*A156919(k,0) for n = 2, 3, .., see A162005. - Johannes W. Meijer, Jun 27 2009
G.f.: 1/(1-1*2*x/(1-2*3*x/(1-3*4*x/(1-4*5*x/(1-5*6*x/(1-... (continued fraction). - Paul Barry, Feb 24 2010
From Paul Barry, Mar 29 2010: (Start)
G.f.: 1/(1-2x-12x^2/(1-18x-240x^2/(1-50x-1260x^2/(1-98x-4032x^2/(1-162x-9900x^2/(1-... (continued fraction);
coefficient sequences given by 4*(n+1)^2*(2n+1)*(2n+3) and 2(2n+1)^2 (see Van Fossen Conrad reference). (End)
E.g.f.: x*Sum_{n>=0} Product_{k=1..n} tanh(2*k*x) = Sum_{n>=1} a(n)*x^n/(n-1)!. - Paul D. Hanna, May 11 2010 [corrected by Paul D. Hanna, Sep 28 2023]
a(n) = (-1)^(n+1)*Sum_{j=1..2*n+1} j!*Stirling2(2*n+1,j)*2^(2*n+1-j)*(-1)^j for n >= 0. Vladimir Kruchinin, Aug 23 2010: (Start)
If n is odd such that 2*n-1 is prime, then a(n) == 1 (mod (2*n-1)); if n is even such that 2*n-1 is prime, then a(n) == -1 (mod (2*n-1)). - Vladimir Shevelev, Sep 01 2010
Recursion: a(n) = (-1)^(n-1) + Sum_{i=1..n-1} (-1)^(n-i+1)*C(2*n-1,2*i-1)* a(i). - Vladimir Shevelev, Aug 08 2011
E.g.f.: tan(x) = Sum_{n>=1} a(n)*x^(2*n-1)/(2*n-1)! = x/(1 - x^2/(3 - x^2/(5 - x^2/(7 - x^2/(9 - x^2/(11 - x^2/(13 -...))))))) (continued fraction from J. H. Lambert - 1761). - Paul D. Hanna, Sep 21 2011
From Sergei N. Gladkovskii, Oct 31 2011 to Oct 09 2013: (Start)
Continued fractions:
E.g.f.: (sec(x))^2 = 1+x^2/(x^2+U(0)) where U(k) = (k+1)*(2k+1) - 2x^2 + 2x^2*(k+1)*(2k+1)/U(k+1).
E.g.f.: tan(x) = x*T(0) where T(k) = 1-x^2/(x^2-(2k+1)*(2k+3)/T(k+1)).
E.g.f.: tan(x) = x/(G(0)+x) where G(k) = 2*k+1 - 2*x + x/(1 + x/G(k+1)).
E.g.f.: tanh(x) = x/(G(0)-x) where G(k) = k+1 + 2*x - 2*x*(k+1)/G(k+1).
E.g.f.: tan(x) = 2*x - x/W(0) where W(k) = 1 + x^2*(4*k+5)/((4*k+1)*(4*k+3)*(4*k+5) - 4*x^2*(4*k+3) + x^2*(4*k+1)/W(k+1)).
E.g.f.: tan(x) = x/T(0) where T(k) = 1 - 4*k^2 + x^2*(1 - 4*k^2)/T(k+1).
E.g.f.: tan(x) = -3*x/(T(0)+3*x^2) where T(k)= 64*k^3 + 48*k^2 - 4*k*(2*x^2 + 1) - 2*x^2 - 3 - x^4*(4*k -1)*(4*k+7)/T(k+1).
G.f.: 1/G(0) where G(k) = 1 - 2*x*(2*k+1)^2 - x^2*(2*k+1)*(2*k+2)^2*(2*k+3)/G(k+1).
G.f.: 2*Q(0) - 1 where Q(k) = 1 + x^2*(4*k + 1)^2/(x + x^2*(4*k + 1)^2 - x^2*(4*k + 3)^2*(x + x^2*(4*k + 1)^2)/(x^2*(4*k + 3)^2 + (x + x^2*(4*k + 3)^2)/Q(k+1) )).
G.f.: (1 - 1/G(0))*sqrt(-x), where G(k) = 1 + sqrt(-x) - x*(k+1)^2/G(k+1).
G.f.: Q(0), where Q(k) = 1 - x*(k+1)*(k+2)/( x*(k+1)*(k+2) - 1/Q(k+1)). (End)
O.g.f.: x + 2*x*Sum_{n>=1} x^n * Product_{k=1..n} (2*k-1)^2 / (1 + (2*k-1)^2*x). - Paul D. Hanna, Feb 05 2013
a(n) = (-4)^n*Li_{1-2*n}(-1). - Peter Luschny, Jun 28 2012
a(n) = (-4)^n*(4^n-1)*Zeta(1-2*n). - Jean-François Alcover, Dec 05 2013
Asymptotic expansion: 4*((2*(2*n-1))/(Pi*e))^(2*n-1/2)*exp(1/2+1/(12*(2*n-1))-1/(360*(2*n-1)^3)+1/(1260*(2*n-1)^5)-...). (See Luschny link.) - Peter Luschny, Jul 14 2015
From Peter Bala, Sep 11 2015: (Start)
The e.g.f. A(x) = tan(x) satisfies the differential equation A''(x) = 2*A(x)*A'(x) with A(0) = 0 and A'(0) = 1, leading to the recurrence a(0) = 0, a(1) = 1, else a(n) = 2*Sum_{i = 0..n-2} binomial(n-2,i)*a(i)*a(n-1-i) for the aerated sequence [0, 1, 0, 2, 0, 16, 0, 272, ...].
Note, the same recurrence, but with the initial conditions a(0) = 1 and a(1) = 1, produces the sequence n! and with a(0) = 1/2 and a(1) = 1 produces A080635. Cf. A002105, A234797. (End)
a(n) = 2*polygamma(2*n-1, 1/2)/Pi^(2*n). - Vladimir Reshetnikov, Oct 18 2015
a(n) = 2^(2n-2)*|p(2n-1,-1/2)|, where p_n(x) are the shifted row polynomials of A019538. E.g., a(2) = 2 = 2^2 * |1 + 6(-1/2) + 6(-1/2)^2|. - Tom Copeland, Oct 19 2016
From Peter Bala, May 05 2017: (Start)
With offset 0, the o.g.f. A(x) = 1 + 2*x + 16*x^2 + 272*x^3 + ... has the property that its 4th binomial transform 1/(1 - 4*x) A(x/(1 - 4*x)) has the S-fraction representation 1/(1 - 6*x/(1 - 2*x/(1 - 20*x/(1 - 12*x/(1 - 42*x/(1 - 30*x/(1 - ...))))))), where the coefficients [6, 2, 20, 12, 42, 30, ...] in the partial numerators of the continued fraction are obtained from the sequence [2, 6, 12, 20, ..., n*(n + 1), ...] by swapping adjacent terms. Compare with the S-fraction associated with A(x) given above by Paul Barry.
A(x) = 1/(1 + x - 3*x/(1 - 4*x/(1 + x - 15*x/(1 - 16*x/(1 + x - 35*x/(1 - 36*x/(1 + x - ...))))))), where the unsigned coefficients in the partial numerators [3, 4, 15, 16, 35, 36,...] come in pairs of the form 4*n^2 - 1, 4*n^2 for n = 1,2,.... (End)
a(n) = Sum_{k = 1..n-1} binomial(2*n-2, 2*k-1) * a(k) * a(n-k), with a(1) = 1. - Michael Somos, Aug 02 2018
a(n) = 2^(2*n-1) * |Euler(2*n-1, 0)|, where Euler(n,x) are the Euler polynomials. - Daniel Suteu, Nov 21 2018 (restatement of one of Copeland's 2007 formulas.)
x - Sum_{n >= 1} (-1)^n*a(n)*x^(2*n)/(2*n)! = x - log(cosh(x)). The series reversion of x - log(cosh(x)) is (1/2)*x - (1/2)*log(2 - exp(x)) = Sum_{n >= 0} A000670(n)*x^(n+1)/(n+1)!. - Peter Bala, Jul 11 2022
For n > 1, a(n) = 2*Sum_{j=1..n-1} Sum_{k=1..j} binomial(2*j,j+k)*(-4*k^2)^(n-1)*(-1)^k/(4^j). - Tani Akinari, Sep 20 2023
a(n) = A110501(n) * 4^(n-1) / n (Han and Liu, 2018). - Amiram Eldar, May 17 2024

A007290 a(n) = 2*binomial(n,3).

Original entry on oeis.org

0, 0, 0, 2, 8, 20, 40, 70, 112, 168, 240, 330, 440, 572, 728, 910, 1120, 1360, 1632, 1938, 2280, 2660, 3080, 3542, 4048, 4600, 5200, 5850, 6552, 7308, 8120, 8990, 9920, 10912, 11968, 13090, 14280, 15540, 16872, 18278, 19760, 21320, 22960, 24682, 26488, 28380, 30360, 32430, 34592, 36848, 39200

Offset: 0

N. J. A. Sloane, Simon Plouffe

Number of acute triangles made from the vertices of a regular n-polygon when n is even (cf. A000330). - Sen-Peng Eu, Apr 05 2001
a(n+2) is (-1)*coefficient of X in Zagier's polynomial (n,n-1). - Benoit Cloitre, Oct 12 2002
Definite integrals of certain products of 2 derivatives of (orthogonal) Chebyshev polynomials of the 2nd kind are pi-multiple of this sequence. For even (p+q): Integrate[ D[ChebyshevU[p, x], x] D[ChebyshevU[q, x], x] (1 - x^2)^(1/2), {x,-1,1}] / Pi = a(n), where n=Min[p,q]. Example: a(3)=20 because Integrate[ D[ChebyshevU[3, x], x] D[ChebyshevU[5, x], x] (1 - x^2)^(1/2), {x,-1,1}]/Pi = 20 since 3=Min[3,5] and 3+5 is even. - Christoph Pacher (Christoph.Pacher(AT)arcs.ac.at), Dec 16 2004
If Y is a 2-subset of an n-set X then, for n>=3, a(n-1) is the number of 3-subsets and 4-subsets of X having exactly one element in common with Y. - Milan Janjic, Dec 28 2007
a(n) is also the number of proper colorings of the cycle graph Csub3 (also the complete graph Ksub3) when n colors are available. - Gary E. Stevens, Dec 28 2008
a(n) is the reverse Wiener index of the path graph with n vertices. See the Balaban et al. reference, p. 927.
For n > 1: a(n) = sum of (n-1)-th row of A141418. - Reinhard Zumkeller, Nov 18 2012
This is the sequence for nuclear magic numbers in an idealized spherical nucleus under the harmonic oscillator model. - Jess Tauber, May 20 2013
Shifted non-vanishing diagonal of A132440^3/3. Second subdiagonal of A238363 (without zeros). For n>0, a(n+2)=n*(n+1)*(n+2)/3. Cf. A130534 for relations to colored forests and disposition of flags on flagpoles. - Tom Copeland, Apr 05 2014
a(n) is the number of ordered rooted trees with n non-root nodes that have 2 leaves; see A108838. - Joerg Arndt, Aug 18 2014
Number of floating point multiplications in the factorization of an (n-1)X(n-1) real matrix by Gaussian elimination as e.g. implemented in LINPACK subroutines sgefa.f or dgefa.f. The number of additions is given by A000330. - Hugo Pfoertner, Mar 28 2018
a(n+1) = Max_{s in S_n} Sum_{k=1..n} (k - s(k))^2 where S_n is the symmetric group of permutations of [1..n]; this maximum is obtained with the permutation s = (1, n) (2, n-1) (3, n-2) ... (k, n-k+1). (see Protat reference). - Bernard Schott, Dec 26 2022

Luigi Berzolari, Allgemeine Theorie der Höheren Ebenen Algebraischen Kurven, Encyclopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen. Band III_2. Heft 3, Leipzig: B. G. Teubner, 1906, p. 352.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 259.
Maurice Protat, Des Olympiades à l'Agrégation, un problème de maximum, Problème 36, p. 83, Ellipses, Paris 1997.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Alexandru T. Balaban, Denise Mills, Ovidiu Ivanciuc and Subhash C. Basak,, Reverse Wiener indices, Croatica Chemica Acta, Vol. 73, No. 4 (2000), pp. 923-941.
A. Burstein, S. Kitaev and T. Mansour, Partially ordered patterns and their combinatorial interpretations, PU. M. A. Vol. 19, No. 2-3 (2008), pp. 27-38.
Otto Haxel, J. Hans D. Jensen and Hans E. Suess, On the "Magic Numbers" in Nuclear Structure, Phys. Rev., Vol. 75 (1949), p. 1766.
Xiangdong Ji, Chapter 8: Structure of Finite Nuclei, Lecture notes for Phys 741 at Univ. of Maryland, p. 140 [From _Tom Copeland_, Apr 07 2014].
Sandi Klavžar, Balázs Patkós, Gregor Rus and Ismael G. Yero, On general position sets in Cartesian grids, arXiv:1907.04535 [math.CO], 2019.
Vladimir Ladma, Magic Numbers.
Cleve Moler, LINPACK subroutine sgefa.f, University of New Mexico, Argonne National Lab, 1978.
Hamzeh Mujahed and Benedek Nagy, Wiener Index on Lines of Unit Cells of the Body-Centered Cubic Grid, Mathematical Morphology and Its Applications to Signal and Image Processing, 12th International Symposium, ISMM 2015.
V. B. Priezzhev, Series expansion for rectilinear polymers on the square lattice, J. Phys. A, Vol. 12, No. 11 (1979), pp. 2131-2139.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Wikipedia, p-derivation.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

A diagonal of A059419. Partial sums of A002378.
A diagonal of A008291. Row 3 of A074650.
Cf. A000292, A051925, A145066, A145067, A145068, A210569.

a007290 n = if n < 3 then 0 else 2 * a007318 n 3  -- Reinhard Zumkeller, Nov 18 2012

I:=[0, 0, 0, 2]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..45]]; // Vincenzo Librandi, Jun 19 2012

A007290 := proc(n) 2*binomial(n,3) end proc:

Table[Integrate[ D[ChebyshevU[n, x], x] D[ChebyshevU[n, x], x] (1 - x^2)^(1/2), {x, -1, 1}]/Pi, {n, 1, 20}] (* Pacher *)
LinearRecurrence[{4,-6,4,-1},{0,0,0,2},50] (* Vincenzo Librandi, Jun 19 2012 *)

my(x='x+O('x^100)); concat([0, 0, 0], Vec(2*x^3/(1-x)^4)) \\ Altug Alkan, Nov 01 2015

apply( {A007290(n)=binomial(n,3)*2}, [0..55]) \\ M. F. Hasler, Jul 02 2021

G.f.: 2*x^3/(1-x)^4.
a(n) = a(n-1)*n/(n-3) = a(n-1) + A002378(n-2) = 2*A000292(n-2) = Sum_{i=0..n-2} i*(i+1) = n*(n-1)*(n-2)/3. - Henry Bottomley, Jun 02 2000 [Formula corrected by R. J. Mathar, Dec 13 2010]
a(n) = A000217(n-2) + A000330(n-2), n>1. - Reinhard Zumkeller, Mar 20 2008
a(n+1) = A000330(n) - A000217(n), n>=0. - Zak Seidov, Aug 07 2010
a(n) = A033487(n-2) - A052149(n-1) for n>1. - Bruno Berselli, Dec 10 2010
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 19 2012
a(n) = (2*n - 3*n^2 + n^3)/3. - T. D. Noe, May 20 2013
a(n+1) = A002412(n) - A000330(n) or "Hex Pyramidal" - "Square Pyramidal" (as can also be seen via above formula). - Richard R. Forberg, Aug 07 2013
Sum_{n>=3} 1/a(n) = 3/4. - Enrique Pérez Herrero, Nov 10 2013
E.g.f.: exp(x)*x^3/3. - Geoffrey Critzer, Nov 22 2015
a(n+2) = delta(-n) = -delta(n) for n >= 0, where delta is the p-derivation over the integers with respect to prime p = 3. - Danny Rorabaugh, Nov 10 2017
(a(n) + a(n+1))/2 = A000330(n-1). - Ezhilarasu Velayutham, Apr 05 2019
Sum_{n>=3} (-1)^(n+1)/a(n) = 6*log(2) - 15/4. - Amiram Eldar, Jan 09 2022
a(n) = Sum_{m=0..n-2} Sum_{k=0..n-2} abs(m-k). - Nicolas Bělohoubek, Nov 06 2022
From Bernard Schott, Jan 04 2023: (Start)
a(n) = 2 * A000292(n-2), for n >= 2.
a(n+1) = 2 *Sum_{k=1..floor(n/2)} (n-(2k-1))^2, for n >= 2. (End)

A155585 a(n) = 2^n*E(n, 1) where E(n, x) are the Euler polynomials.

Original entry on oeis.org

1, 1, 0, -2, 0, 16, 0, -272, 0, 7936, 0, -353792, 0, 22368256, 0, -1903757312, 0, 209865342976, 0, -29088885112832, 0, 4951498053124096, 0, -1015423886506852352, 0, 246921480190207983616, 0, -70251601603943959887872, 0, 23119184187809597841473536, 0

Offset: 0

Paul D. Hanna, Jan 24 2009

Previous name was: a(n) = Sum_{k=0..n-1} (-1)^(k)*C(n-1,k)*a(n-1-k)*a(k) for n>0 with a(0)=1.
Factorials have a similar recurrence: f(n) = Sum_{k=0..n-1} C(n-1,k)*f(n-1-k)*f(k), n > 0.
Related to A102573: letting T(q,r) be the coefficient of n^(r+1) in the polynomial 2^(q-n)/n times Sum_{k=0..n} binomial(n,k)*k^q, then A155585(x) = Sum_{k=0..x-1} T(x,k)*(-1)^k. See Mathematica code below. - John M. Campbell, Nov 16 2011
For the difference table and the relation to the Seidel triangle see A239005. - Paul Curtz, Mar 06 2014
From Tom Copeland, Sep 29 2015: (Start)
Let z(t) = 2/(e^(2t)+1) = 1 + tanh(-t) = e.g.f.(-t) for this sequence = 1 - t + 2*t^3/3! - 16*t^5/5! + ... .
dlog(z(t))/dt = -z(-t), so the raising operators that generate Appell polynomials associated with this sequence, A081733, and its reciprocal, A119468, contain z(-d/dx) = e.g.f.(d/dx) as the differential operator component.
dz(t)/dt = z*(z-2), so the assorted relations to a Ricatti equation, the Eulerian numbers A008292, and the Bernoulli numbers in the Rzadkowski link hold.
From Michael Somos's formula below (drawing on the Edwards link), y(t,1)=1 and x(t,1) = (1-e^(2t))/(1+e^(2t)), giving z(t) = 1 + x(t,1). Compare this to the formulas in my list in A008292 (Sep 14 2014) with a=1 and b=-1,
A) A(t,1,-1) = A(t) = -x(t,1) = (e^(2t)-1)/(1+e^(2t)) = tanh(t) = t + -2*t^3/3! + 16*t^5/5! + -272*t^7/7! + ... = e.g.f.(t) - 1 (see A000182 and A000111)
B) Ainv(t) = log((1+t)/(1-t))/2 = tanh^(-1)(t) = t + t^3/3 + t^5/5 + ..., the compositional inverse of A(t)
C) dA/dt = (1-A^2), relating A(t) to a Weierstrass elliptic function
D) ((1-t^2)d/dt)^n t evaluated at t=0, a generator for the sequence A(t)
F) FGL(x,y)= (x+y)/(1+xy) = A(Ainv(x) + Ainv(y)), a related formal group law corresponding to the Lorentz FGL (Lorentz transformation--addition of parallel velocities in special relativity) and the Atiyah-Singer signature and the elliptic curve (1-t^2)*s = t^3 in Tate coordinates according to the Lenart and Zainoulline link and the Buchstaber and Bunkova link (pp. 35-37) in A008292.
A133437 maps the reciprocal odd natural numbers through the refined faces of associahedra to a(n).
A145271 links the differential relations to the geometry of flow maps, vector fields, and thereby formal group laws. See Mathworld for links of tanh to other geometries and statistics.
Since the a(n) are related to normalized values of the Bernoulli numbers and the Riemann zeta and Dirichlet eta functions, there are links to Witten's work on volumes of manifolds in two-dimensional quantum gauge theories and the Kervaire-Milnor formula for homotopy groups of hyperspheres (see my link below).
See A101343, A111593 and A059419 for this and the related generator (1 + t^2) d/dt and associated polynomials. (End)
With the exception of the first term (1), entries are the alternating sums of the rows of the Eulerian triangle, A008292. - Gregory Gerard Wojnar, Sep 29 2018

			E.g.f.: 1 + x - 2*x^3/3! + 16*x^5/5! - 272*x^7/7! + 7936*x^9/9! -+ ... = exp(x)/cosh(x).
O.g.f.: 1 + x - 2*x^3 + 16*x^5 - 272*x^7 + 7936*x^9 - 353792*x^11 +- ...
O.g.f.: 1 + x/(1+2*x) + 2!*x^2/((1+2*x)*(1+4*x)) + 3!*x^3/((1+2*x)*(1+4*x)*(1+6*x)) + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
P. Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 28.
Tom Copeland, The Elliptic Lie Triad: Ricatti and KdV Equations, Infinigens, and Elliptic Genera
Tom Copeland, The Kervaire-Milnor formula
H. M. Edwards, A normal form for elliptic curves, Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 3, 393-422. see section 12, pp. 410-411.
Wenjie Fang, Hsien-Kuei Hwang, and Mihyun Kang, Phase transitions from exp(n^(1/2)) to exp(n^(2/3)) in the asymptotics of banded plane partitions, arXiv:2004.08901 [math.CO], 2020.
G. Rzadkowski, Bernoulli numbers and solitons-revisited, Journal of Nonlinear Math. Physics, 1711, pp. 121-126.

Cf. A001057, A009006, A102573.
Equals row sums of A119879. - Johannes W. Meijer, Apr 20 2011
Cf. A081733, A119468, A008292, A000182, A000111, A101343, A111593, A059419, A133437, A145271.
(-1)^n*a(n) are the alternating row sums of A123125. - Wolfdieter Lang, Jul 12 2017

A155585 := n -> 2^n*euler(n, 1): # Peter Luschny, Jan 26 2009
a := proc(n) option remember; `if`(n::even, 0^n, -(-1)^n - add((-1)^k*binomial(n,k) *a(n-k), k = 1..n-1)) end: # Peter Luschny, Jun 01 2016
# Or via the recurrence of the Fubini polynomials:
F := proc(n) option remember; if n = 0 then return 1 fi;
expand(add(binomial(n, k)*F(n-k)*x, k = 1..n)) end:
a := n -> (-2)^n*subs(x = -1/2, F(n)):
seq(a(n), n = 0..30); # Peter Luschny, May 21 2021

a[m_] := Sum[(-2)^(m - k) k! StirlingS2[m, k], {k, 0, m}] (* Peter Luschny, Apr 29 2009 *)
poly[q_] :=  2^(q-n)/n*FunctionExpand[Sum[Binomial[n, k]*k^q, {k, 0, n}]]; T[q_, r_] :=  First[Take[CoefficientList[poly[q], n], {r+1, r+1}]]; Table[Sum[T[x, k]*(-1)^k, {k, 0, x-1}], {x, 1, 16}] (* John M. Campbell, Nov 16 2011 *)
f[n_] := (-1)^n 2^(n+1) PolyLog[-n, -1]; f[0] = -f[0]; Array[f, 27, 0] (* Robert G. Wilson v, Jun 28 2012 *)

a(n)=if(n==0,1,sum(k=0,n-1,(-1)^(k)*binomial(n-1,k)*a(n-1-k)*a(k)))

a(n)=local(X=x+x*O(x^n));n!*polcoeff(exp(X)/cosh(X),n)

a(n)=polcoeff(sum(m=0,n,m!*x^m/prod(k=1,m,1+2*k*x+x*O(x^n))),n) \\ Paul D. Hanna, Jul 20 2011

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); n! * polcoeff( 1 + sinh(x + A) / cosh(x + A), n))} /* Michael Somos, Jan 16 2012 */

a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,intformal(subst(A,x,-x)+x*O(x^n))^k/k!));n!*polcoeff(A,n)
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Nov 25 2013

from sympy import bernoulli
def A155585(n): return (((2<<(m:=n+1))-2)*bernoulli(m)<>1) if n&1 else (0 if n else 1) # Chai Wah Wu, Apr 14 2023

def A155585(n) :
    if n == 0 : return 1
    return add(add((-1)^(j+1)*binomial(n+1,k-j)*j^n for j in (0..k)) for k in (1..n))
[A155585(n) for n in (0..26)] # Peter Luschny, Jul 23 2012

def A155585_list(n): # Akiyama-Tanigawa algorithm
    A = [0]*(n+1); R = []
    for m in range(n+1) :
        d = divmod(m+3, 4)
        A[m] = 0 if d[1] == 0 else (-1)^d[0]/2^(m//2)
        for j in range(m, 0, -1) :
            A[j - 1] = j * (A[j - 1] - A[j])
        R.append(A[0])
    return R
A155585_list(30) # Peter Luschny, Mar 09 2014

E.g.f.: exp(x)*sech(x) = exp(x)/cosh(x). (See A009006.) - Paul Barry, Mar 15 2006
Sequence of absolute values is A009006 (e.g.f. 1+tan(x)).
O.g.f.: Sum_{n>=0} n! * x^n / Product_{k=1..n} (1 + 2*k*x). - Paul D. Hanna, Jul 20 2011
a(n) = 2^n*E_{n}(1) where E_{n}(x) are the Euler polynomials. - Peter Luschny, Jan 26 2009
a(n) = EL_{n}(-1) where EL_{n}(x) are the Eulerian polynomials. - Peter Luschny, Aug 03 2010
a(n+1) = (4^n-2^n)*B_n(1)/n, where B_{n}(x) are the Bernoulli polynomials (B_n(1) = B_n for n <> 1). - Peter Luschny, Apr 22 2009
G.f.: 1/(1-x+x^2/(1-x+4*x^2/(1-x+9*x^2/(1-x+16*x^2/(1-...))))) (continued fraction). - Paul Barry, Mar 30 2010
G.f.: -log(x/(exp(x)-1))/x = Sum_{n>=0} a(n)*x^n/(2^(n+1)*(2^(n+1)-1)*n!). - Vladimir Kruchinin, Nov 05 2011
E.g.f.: exp(x)/cosh(x) = 2/(1+exp(-2*x)) = 2/(G(0) + 1); G(k) = 1 - 2*x/(2*k + 1 - x*(2*k+1)/(x - (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 10 2011
E.g.f. is x(t,1) + y(t,1) where x(t,a) and y(t,a) satisfy y(t,a)^2 = (a^2 - x(t,a)^2) / (1 - a^2 * x(t,a)^2) and dx(t,a) / dt = y(t,a) * (1 - a * x(t,a)^2) and are the elliptic functions of Edwards. - Michael Somos, Jan 16 2012
E.g.f.: 1/(1 - x/(1+x/(1 - x/(3+x/(1 - x/(5+x/(1 - x/(7+x/(1 - x/(9+x/(1 +...))))))))))), a continued fraction. - Paul D. Hanna, Feb 11 2012
E.g.f. satisfies: A(x) = Sum_{n>=0} Integral( A(-x) dx )^n / n!. - Paul D. Hanna, Nov 25 2013
a(n) = -2^(n+1)*Li_{-n}(-1). - Peter Luschny, Jun 28 2012
a(n) = Sum_{k=1..n} Sum_{j=0..k} (-1)^(j+1)*binomial(n+1,k-j)*j^n for n > 0. - Peter Luschny, Jul 23 2012
From Sergei N. Gladkovskii, Oct 25 2012 to Dec 16 2013: (Start)
Continued fractions:
G.f.: 1 + x/T(0) where T(k) = 1 + (k+1)*(k+2)*x^2/T(k+1).
E.g.f.: exp(x)/cosh(x) = 1 + x/S(0) where S(k) = (2*k+1) + x^2/S(k+1).
E.g.f.: 1 + x/(U(0)+x) where U(k) = 4*k+1 - x/(1 + x/(4*k+3 - x/(1 + x/U(k+1)))).
E.g.f.: 1 + tanh(x) = 4*x/(G(0)+2*x) where G(k) = 1 - (k+1)/(1 - 2*x/(2*x + (k+1)^2/G(k+1)));
G.f.: 1 + x/G(0) where G(k) = 1 + 2*x^2*(2*k+1)^2 - x^4*(2*k+1)*(2*k+2)^2*(2*k+3)/G(k+1) (due to Stieltjes).
E.g.f.: 1 + x/(G(0) + x) where G(k) = 1 - 2*x/(1 + (k+1)/G(k+1)).
G.f.: 2 - 1/Q(0) where Q(k) = 1 + x*(k+1)/( 1 - x*(k+1)/Q(k+1)).
G.f.: 2 - 1/Q(0) where Q(k) = 1 + x*k^2 + x/(1 - x*(k+1)^2/Q(k+1)).
G.f.: 1/Q(0) where Q(k) = 1 - 2*x + x*(k+1)/(1-x*(k+1)/Q(k+1)).
G.f.: 1/Q(0) where Q(k) = 1 - x*(k+1)/(1 + x*(k+1)/Q(k+1)).
E.g.f.: 1 + x*Q(0) where Q(k) = 1 - x^2/( x^2 + (2*k+1)*(2*k+3)/Q(k+1)).
G.f.: 2 - T(0)/(1+x) where T(k) = 1 - x^2*(k+1)^2/(x^2*(k+1)^2 + (1+x)^2/T(k+1)).
E.g.f.: 1/(x - Q(0)) where Q(k) = 4*k^2 - 1 + 2*x + x^2*(2*k-1)*(2*k+3)/Q(k+1). (End)
G.f.: 1 / (1 - b(1)*x / (1 - b(2)*x / (1 - b(3)*x / ... ))) where b = A001057. - Michael Somos, Jan 03 2013
From Paul Curtz, Mar 06 2014: (Start)
a(2n) = A000007(n).
a(2n+1) = (-1)^n*A000182(n+1).
a(n) is the binomial transform of A122045(n).
a(n) is the row sum of A081658. For fractional Euler numbers see A238800.
a(n) + A122045(n) = 2, 1, -1, -2, 5, 16, ... = -A163982(n).
a(n) - A122045(n) = -A163747(n).
a(n) is the Akiyama-Tanigawa transform applied to 1, 0, -1/2, -1/2, -1/4, 0, ... = A046978(n+3)/A016116(n). (End)
a(n) = 2^(2*n+1)*(zeta(-n,1/2) - zeta(-n, 1)), where zeta(a, z) is the generalized Riemann zeta function. - Peter Luschny, Mar 11 2015
a(n) = 2^(n + 1)*(2^(n + 1) - 1)*Bernoulli(n + 1, 1)/(n + 1). (From Bill Gosper, Oct 28 2015) - N. J. A. Sloane, Oct 28 2015 [See the above comment from Peter Luschny, Apr 22 2009.]
a(n) = -(n mod 2)*((-1)^n + Sum_{k=1..n-1} (-1)^k*C(n,k)*a(n-k)) for n >= 1. - Peter Luschny, Jun 01 2016
a(n) = (-2)^n*F_{n}(-1/2), where F_{n}(x) is the Fubini polynomial. - Peter Luschny, May 21 2021

New name from Peter Luschny, Mar 12 2015

A006229 Expansion of e.g.f. exp( tan x ).

Original entry on oeis.org

1, 1, 1, 3, 9, 37, 177, 959, 6097, 41641, 325249, 2693691, 24807321, 241586893, 2558036145, 28607094455, 342232522657, 4315903789009, 57569080467073, 807258131578995, 11879658510739497, 183184249105857781, 2948163649552594737, 49548882107764546223

Offset: 0

N. J. A. Sloane, Jeffrey Shallit

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 259, Sum_{k} T(n,k).
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.
L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 150.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..480 (terms 0..100 from T. D. Noe)
J. Shallit, Letter to N. J. A. Sloane, May 1975
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Row sums of A059419 and unsigned A111593.

Cf. A003711, A003717, A000182, A047691, A047692.

function A006229_list(len::Int)
    len <= 0 && return BigInt[]
    T = zeros(BigInt, len, len); T[1,1] = 1
    S = Array(BigInt, len); S[1] = 1
    for n in 2:len
        T[n,n] = 1
        for k in 2:n-1 T[n,k] = T[n-1,k-1] + k*(k-1)*T[n-1,k+1] end
        S[n] = sum(T[n,k] for k in 2:n)
    end
S end
println(A006229_list(24)) # Peter Luschny, Apr 27 2017

With[{nn=30},CoefficientList[Series[Exp[Tan[x]],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Oct 04 2011 *)

a(n):=sum(if oddp(n+k) then 0 else (-1)^((n+k)/2)*sum(j!/k!*stirling2(n,j)*2^(n-j)*(-1)^(n+j-k)*binomial(j-1, k-1), j, k, n), k, 1, n); /* Vladimir Kruchinin, Aug 05 2010 */

E.g.f.: exp(tan(x)).
a(n) = sum(if oddp(n+k) then 0 else (-1)^((n+k)/2)*sum(j!/k!*stirling2(n,j)*2^(n-j)*(-1)^(n+j-k)*binomial(j-1,k-1),j,k,n),k,1,n), n>0. - Vladimir Kruchinin, Aug 05 2010
E.g.f.: 1 + tan(x)/T(0), where T(k) = 4*k+1 - tan(x)/(2 + tan(x)/(4*k+3 - tan(x)/(2 + tan(x)/T(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Dec 03 2013
a(n) = Sum_{i=0..(n-1)/2} binomial(n-1,2*i)*z(i+1)*a(n-2*i-1), a(0)=1, where z(n) is tangent (or "zag") numbers (A000182). - Vladimir Kruchinin, Mar 04 2015 [corrected by Jason Yuen, Dec 29 2024]

More terms from Larry Reeves (larryr(AT)acm.org), Feb 08 2001

A185896 Triangle of coefficients of (1/sec^2(x))*D^n(sec^2(x)) in powers of t = tan(x), where D = d/dx.

Original entry on oeis.org

1, 0, 2, 2, 0, 6, 0, 16, 0, 24, 16, 0, 120, 0, 120, 0, 272, 0, 960, 0, 720, 272, 0, 3696, 0, 8400, 0, 5040, 0, 7936, 0, 48384, 0, 80640, 0, 40320, 7936, 0, 168960, 0, 645120, 0, 846720, 0, 362880, 0, 353792, 0, 3256320, 0, 8951040, 0, 9676800, 0, 3628800

Offset: 0

Peter Bala, Feb 07 2011

DEFINITION
Define polynomials R(n,t) with t = tan(x) by
... (d/dx)^n sec^2(x) = R(n,tan(x))*sec^2(x).
The first few are
... R(0,t) = 1
... R(1,t) = 2*t
... R(2,t) = 2 + 6*t^2
... R(3,t) = 16*t + 24*t^3.
This triangle shows the coefficients of R(n,t) in ascending powers of t called the tangent number triangle in [Hodges and Sukumar].
The polynomials R(n,t) form a companion polynomial sequence to Hoffman's two polynomial sequences - P(n,t) (A155100), the derivative polynomials of the tangent and Q(n,t) (A104035), the derivative polynomials of the secant. See also A008293 and A008294.
COMBINATORIAL INTERPRETATION
A combinatorial interpretation for the polynomial R(n,t) as the generating function for a sign change statistic on certain types of signed permutation can be found in [Verges].
A signed permutation is a sequence (x_1,x_2,...,x_n) of integers such that {|x_1|,|x_2|,...|x_n|} = {1,2...,n}. They form a group, the hyperoctahedral group of order 2^n*n! = A000165(n), isomorphic to the group of symmetries of the n dimensional cube.
Let x_1,...,x_n be a signed permutation.
Then 0,x_1,...,x_n,0 is a snake of type S(n;0,0) when 0 < x_1 > x_2 < ... 0.
For example, 0 4 -3 -1 -2 0 is a snake of type S(4;0,0).
Let sc be the number of sign changes through a snake
... sc = #{i, 1 <= i <= n-1, x_i*x_(i+1) < 0}.
For example, the snake 0 4 -3 -1 -2 0  has sc = 1. The polynomial R(n,t) is the generating function for the sign change statistic on snakes of type S(n+1;0,0):
... R(n,t) = sum {snakes in S(n+1;0,0)} t^sc.
See the example section below for the cases n=1 and n=2.
PRODUCTION MATRIX
Define three arrays R, L, and S as
... R = superdiag[2,3,4,...]
... L = subdiag[1,2,3,...]
... S = diag[2,4,6,...]
with the indicated sequences on the main superdiagonal, the main subdiagonal and main diagonal, respectively, and 0's elsewhere. The array R+L is the production array for this triangle: the first row of (R+L)^n produces the n-th row of the triangle.
On the vector space of complex polynomials the array R, the raising operator, represents the operator p(x) - > d/dx (x^2*p(x)), and the array L, the lowering operator, represents the differential operator d/dx - see Formula (4) below.
The three arrays satisfy the commutation relations
... [R,L] = S, [R,S] = 2*R, [L,S] = -2*L
and hence give a representation of the Lie algebra sl(2).

			Table begins
  n\k|.....0.....1.....2.....3.....4.....5.....6
  ==============================================
  0..|.....1
  1..|.....0.....2
  2..|.....2.....0.....6
  3..|.....0....16.....0....24
  4..|....16.....0...120.....0...120
  5..|.....0...272.....0...960.....0...720
  6..|...272.....0..3696.....0..8400.....0..5040
Examples of recurrence relation
  T(4,2) = 3*(T(3,1) + T(3,3)) = 3*(16 + 24) = 120;
  T(6,4) = 5*(T(5,3) + T(5,5)) = 5*(960 + 720) = 8400.
Example of integral formula (6)
... Integral_{t = -1..1} (1-t^2)*(16-120*t^2+120*t^4)*(272-3696*t^2+8400*t^4-5040*t^6) dt = 2830336/1365 = -2^13*Bernoulli(12).
Examples of sign change statistic sc on snakes of type (0,0)
= = = = = = = = = = = = = = = = = = = = = =
.....Snakes....# sign changes sc.......t^sc
= = = = = = = = = = = = = = = = = = = = = =
n=1
...0 1 -2 0...........1................t
...0 2 -1 0...........1................t
yields R(1,t) = 2*t;
n=2
...0 1 -2 3 0.........2................t^2
...0 1 -3 2 0.........2................t^2
...0 2 1 3 0..........0................1
...0 2 -1 3 0.........2................t^2
...0 2 -3 1 0.........2................t^2
...0 3 1 2 0..........0................1
...0 3 -1 2 0.........2................t^2
...0 3 -2 1 0.........2................t^2
yields
R(2,t) = 2 + 6*t^2.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
K. Boyadzhiev, Derivative Polynomials for tanh, tan, sech and sec in Explicit Form, arXiv:0903.0117 [math.CA], 2009-2010.
M-P. Grosset and A. P. Veselov, Bernoulli numbers and solitons, arXiv:math/0503175 [math.GM], 2005.
A. Hodges and C. V. Sukumar, Bernoulli, Euler, permutations and quantum algebras, Proc. R. Soc. A (2007) 463, 2401-2414 doi:10.1098/rspa.2007.0001
Michael E. Hoffman, Derivative polynomials, Euler polynomials,and associated integer sequences, Electronic Journal of Combinatorics, Volume 6 (1999), Research Paper #R21.
Michael E. Hoffman, Derivative polynomials for tangent and secant, Amer. Math. Monthly, 102 (1995), 23-30.
M. Josuat-Verges, Enumeration of snakes and cycle-alternating permutations, arXiv:1011.0929 [math.CO], 2010.
Shi-Mei Ma, Qi Fang, Toufik Mansour, Yeong-Nan Yeh, Alternating Eulerian polynomials and left peak polynomials, arXiv:2104.09374, 2021

Cf. A000182, A000364, A000828 (row sums), A008292, A008293, A008294, A104035, A155100.

R = proc(n) option remember;
if n=0 then RETURN(1);
else RETURN(expand(diff((u^2+1)*R(n-1), u))); fi;
end proc;
for n from 0 to 12 do
t1 := series(R(n), u, 20);
lprint(seriestolist(t1));
od:

Table[(-1)^(n + 1)*(-1)^((n - k)/2)*Sum[j!*StirlingS2[n + 1, j]*2^(n + 1 - j)*(-1)^(n + j - k)*Binomial[j - 1, k], {j, k + 1, n + 1}], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Jul 22 2017 *)

{T(n, k) = if( n<0 || k<0 || k>n, 0, if(n==k, n!, (k+1)*(T(n-1, k-1) + T(n-1, k+1))))};

{T(n, k) = my(A); if( n<0 || k>n, 0, A=1; for(i=1, n, A = ((1 + x^2) * A)'); polcoeff(A, k))}; /* Michael Somos, Jun 24 2017 */

GENERATING FUNCTION
E.g.f.:
(1)... F(t,z) = 1/(cos(z)-t*sin(z))^2 = Sum_{n>=0} R(n,t)*z^n/n! = 1 + (2*t)*z + (2+6*t^2)*z^2/2! + (16*t+24*t^3)*z^3/3! + ....
The e.g.f. equals the square of the e.g.f. of A104035.
Continued fraction representation for the o.g.f:
(2)... F(t,z) = 1/(1-2*t*z - 2*(1+t^2)*z^2/(1-4*t*z -...- n*(n+1)*(1+t^2)*z^2/(1-2*n*(n+1)*t*z -....
RECURRENCE RELATION
(3)... T(n,k) = (k+1)*(T(n-1,k-1) + T(n-1,k+1)).
ROW POLYNOMIALS
The polynomials R(n,t) satisfy the recurrence relation
(4)... R(n+1,t) = d/dt{(1+t^2)*R(n,t)} with R(0,t) = 1.
Let D be the derivative operator d/dt and U = t, the shift operator.
(5)... R(n,t) = (D + DUU)^n 1
RELATION WITH OTHER SEQUENCES
A) Derivative Polynomials A155100
The polynomials (1+t^2)*R(n,t) are the polynomials P_(n+2)(t) of A155100.
B) Bernoulli Numbers A000367 and A002445
Put S(n,t) = R(n,i*t), where i = sqrt(-1). We have the definite integral evaluation
(6)... Integral_{t = -1..1} (1-t^2)*S(m,t)*S(n,t) dt = (-1)^((m-n)/2)*2^(m+n+3)*Bernoulli(m+n+2).
The case m = n is equivalent to the result of [Grosset and Veselov]. The methods used there extend to the general case.
C) Zigzag Numbers A000111
(7)... R_n(1) = A000828(n+1) = 2^n*A000111(n+1).
D) Eulerian Numbers A008292
The polynomials R(n,t) are related to the Eulerian polynomials A(n,t) via
(8)... R(n,t) = (t+i)^n*A(n+1,(t-i)/(t+i))
with the inverse identity
(9)... A(n+1,t) = (-i/2)^n*(1-t)^n*R(n,i*(1+t)/(1-t)),
where {A(n,t)}n>=1 = [1,1+t,1+4*t+t^2,1+11*t+11*t^2+t^3,...] is the sequence of Eulerian polynomials and i = sqrt(-1).
E) Ordered set partitions A019538
(10)... R(n,t) = (-2*i)^n*T(n+1,x)/x,
where x = i/2*t - 1/2 and T(n,x) is the n-th row po1ynomial of A019538;
F) Miscellaneous
Column 1 is the sequence of tangent numbers - see A000182.
A000670(n+1) = (-i/2)^n*R(n,3*i).
A004123(n+2) = 2*(-i/2)^n*R(n,5*i).
A080795(n+1) =(-1)^n*(sqrt(-2))^n*R(n,sqrt(-2)). - Peter Bala, Aug 26 2011
From Leonid Bedratyuk, Aug 12 2012: (Start)
T(n,k) = (-1)^(n+1)*(-1)^((n-k)/2)*Sum_{j=k+1..n+1} j! *stirling2(n+1,j) *2^(n+1-j) *(-1)^(n+j-k) *binomial(j-1,k), see A059419.
Sum_{j=i+1..n+1}((1-(-1)^(j-i))/(2*(j-i))*(-1)^((n-j)/2)*T(n,j))=(n+1)*(-1)^((n-1-i)/2)*T(n-1,i), for n>1 and  0G.f.:  1/G(0,t,x), where G(k,t,x) = 1 - 2*t*x - 2*k*t*x - (1+t^2)*(k+2)*(k+1)*x^2/G(k+1,t,x); (continued fraction due to T. J. Stieltjes). - Sergei N. Gladkovskii, Dec 27 2013

A209849 Triangle read by rows: coefficients of polynomials in Sum_{k = 0..t} k^n * binomial(t,k).

Original entry on oeis.org

1, 1, 1, 0, 3, 1, -2, 3, 6, 1, 0, -10, 15, 10, 1, 16, -30, -15, 45, 15, 1, 0, 112, -210, 35, 105, 21, 1, -272, 588, 28, -735, 280, 210, 28, 1, 0, -2448, 5292, -2436, -1575, 1008, 378, 36, 1, 7936, -18960, 4140, 20160, -14595, -1575, 2730, 630, 45, 1

Offset: 1

Peter Bala, Mar 15 2012

Repeatedly applying the operator x*d/dx to (1 + x)^t (t a nonnegative integer) and evaluating at x = 1 yields Sum_{k = 0..t} k^n*binomial(t,k) = R(n,t)*2^(t-n), where R(n,t) is a polynomial in t for n = 1,2,.... The polynomial sequence {R(n,t)}_{n>=0} is of binomial type. The first few values are given in the example section below.
This triangle lists the coefficients of these polynomials in ascending powers of t (omitting R(0,t) = 1). A closely related triangle is A102573, which lists the coefficients of the polynomials R(n,t) after factors of t and t*(1 + t) have been removed.
This is the case m = 2 of a family of binomial type polynomials satisfying the recurrence R(n+1,t) = t*(m*(R(n,t) - R(n,t-1)) + R(n,t-1)) with R(0,t) = 1. Case m = 0 gives the falling factorials (A008275); Case m = -1 gives a signed version of A079641.

			Repeatedly applying the operator x*d/dx to (1 + x)^n and evaluating the result at x = 1 yields
Sum_{k = 0..n} k   * binomial(n,k) =  n                * 2^(n-1).
Sum_{k = 0..n} k^2 * binomial(n,k) = (n +   n^2)       * 2^(n-2).
Sum_{k = 0..n} k^3 * binomial(n,k) = (    3*n^2 + n^3) * 2^(n-3).
Triangle begins:
  n\k|    1     2     3     4     5     6     7     8
  = = = = = = = = = = = = = = = = = = = = = = = = = =
  1  |    1
  2  |    1     1
  3  |    0     3     1
  4  |   -2     3     6     1
  5  |    0   -10    15    10     1
  6  |   16   -30   -15    45    15     1
  7  |    0   112  -210    35   105    21     1
  8  | -272   588    28  -735   280   210    28     1
  ...

Seiichi Manyama, Rows n = 1..140, flattened

Columns k=1..3 give A155585(n-1), A383165, A383166.

Cf. A008275, A009006 (alt. row sums), A059419, A063376, A075497, A079641, A102573, A119468, A129062, A154602, A176668, A195204, A383140.

# The function BellMatrix is defined in A264428.
g := n -> 2^n*euler(n,1): BellMatrix(g, 9); # Peter Luschny, Jan 21 2016

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[2^# EulerE[#, 1]&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

T(n, k) = sum(j=k, n, 2^(n-j)*stirling(n, j, 2)*stirling(j, k, 1)); \\ Seiichi Manyama, Apr 16 2025

# uses[bell_matrix from A264428]
g = lambda n: sum((-2)^(n-k)*factorial(k)*stirling_number2(n,k) for k in (0..n))
bell_matrix(g, 9) # Peter Luschny, Jan 21 2016

def a_row(n):
    s = sum(2^(n-k)*stirling_number2(n, k)*falling_factorial(x, k) for k in (0..n))
    return expand(s).list()[1:]
for n in (1..10): print(a_row(n)) # Seiichi Manyama, Apr 16 2025

T(n,k) = Sum_{j = 0..n} (-1)^(n+k) * (-2)^(n-j) * Stirling2(n,j) * |Stirling1(j,k)|. [corrected by Seiichi Manyama, Apr 16 2025]
E.g.f.: F(x,t) := (1/2 + 1/2*exp(2*x))^t = (1 + tanh(-x))^(-t) = 1 + t*x + (t+t^2)*x^2/2! + (3*t^2+t^3)*x^3/3! + ... satisfies the delay differential equation d/dx(F(x,t)) = 2*F(x,t) - F(x,t-1).
Recurrence for row polynomials R(n,t): R(n+1,t) = t*(2*R(n,t) - R(n,t-1)) with R(0,t) = 1.
Let D be the backward difference operator D(f(x)) = f(x) - f(x-1). Then (x*D)^n(2^x) = 2^(x-n)*R(n,x). Cf. A079641.
Discrete Dobinski-type relation: R(n,x) = 1/2^x*Sum_{k = 0..inf} (2*k)^n*x*(x - 1)*...*(x - k + 1)/k!, valid for x = 0,1,2,.... and n >= 1.
Other Dobinski-type relations: exp(-x)*Sum_{k = 0..inf} R(n,k)*x^k/k! = n-th row polynomial of A075497.
exp(-x)*Sum_{k = 0..inf} R(n,k+1)*x^k/k! = n-th row polynomial of A154602.
i^(-n)*exp(i*x)*Sum_{k = 0..inf} R(n,-k)*(-i*x)^k/k! = n-th row polynomial of A059419 where i = sqrt(-1).
Writing x^[n] in place of R(n,x) we have the analog of the Bernoulli summation formula for powers of integers: Sum_{k = 1..n-1} k^[p] = 1/(p + 1)*Sum_{k = 0..p} 2^k*binomial(p+1,k)*B_k*n^[p+1-k], where B_k = [1,-1/2,1/6,0,-1/30,...] is the sequence of Bernoulli numbers.
n-th row sum R(n,1) equals 2^(n-1). Alternating row sums R(n,-1) starting [-1,0,2,0,-16,0,272,...] are signed tangent numbers - see A009006 and A155585.
R(n+1,2) = 2^n + 4^n = A063376(n).
Triangle as a product of lower triangular arrays equals A075497*A008275.
The triangle of connection constants between the polynomials (x + 1)^[n] and x^[n] appears to be A119468 = (P^2 + 1)/2, where P denotes Pascal's triangle.
Also the Bell transform of the sequence 2^n*E(n,1), E(n,x) the Euler polynomials (A155585). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 21 2016
From Peter Bala, Jun 26 2016: (Start)
With row and column numbering starting at 0:
E.g.f. is exp(x)/cosh(x)*((1 + exp(2*x))/2)^t = 1 + (1 + t)*x + (3*t + t^2)*x^2/2! + (-2 + 3*t + 6*t^2 + t^3)*x^3/3! + ....
Exponential Riordan array [d/dx(f(x)), f(x)] belonging to the Derivative subgroup of the Riordan group, where f(x) = log((1 + exp(2*x))/2) and df/dx = exp(x)/cosh(x) is the e.g.f. for A155585. (End)
T(n,k) = [x^k] Sum_{k=0..n} 2^(n-k) * Stirling2(n,k) * FallingFactorial(x,k). - Seiichi Manyama, Apr 16 2025
E.g.f. of column k (with leading zeros): f(x)^k / k! with f(x) = log(1 + (exp(2*x) - 1)/2). - Seiichi Manyama, Apr 18 2025

A024283 E.g.f. (1/2) * tan(x)^2 (even powers only).

Original entry on oeis.org

0, 1, 8, 136, 3968, 176896, 11184128, 951878656, 104932671488, 14544442556416, 2475749026562048, 507711943253426176, 123460740095103991808, 35125800801971979943936, 11559592093904798920736768, 4356981378562584648085405696, 1864703851860264785548754812928

Offset: 0

N. J. A. Sloane. This sequence was in the 1973 "Handbook", but was then omitted from the database. Resubmitted by R. H. Hardin. Entry revised by N. J. A. Sloane, Jun 12 2012

Number of cyclically reverse alternating permutations of length 2n+2, cf. A024255. - Vladeta Jovovic, May 20 2007 [Comment corrected by Fausto A. C. Cariboni, Sep 02 2020]

Related to A102573: letting T(q,r) be the coefficient of n^r in the polynomial 2^(q-n)/n times sum(k=0..n binomial(n, k)*k^q), then A024283(x) = sum(k=0..(2*x-1) T(2*x,k)*(-1)^(k+x)*2^k). See Mathematica code below. [John M. Campbell, Sep 15 2013]

			(tan x)^2 = x^2 + 2/3*x^4 + 17/45*x^6 + 62/315*x^8 + ...
G.f. = x + 8*x^2 + 136*x^3 + 3968*x^4 + 176896*x^5 + 11184128*x^6 + ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 259, T(n,2).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..100
Beáta Bényi, Miguel Méndez, José L. Ramírez and Tanay Wakhare, Restricted r-Stirling Numbers and their Combinatorial Applications, arXiv:1811.12897 [math.CO], 2018.

Cf. A009764, A024255.

Cf. A000182, A102573. A diagonal of A059419.

A024283 := n -> `if`(n=0,0,(-1)^(n-1)*2^(2*n+1)*polylog(-2*n-1,-1)); # Peter Luschny, Jun 28 2012

f[n_] := -(-1)^n 2^(2 n + 1) PolyLog[-1 - 2 n, -1]; f[0] = 0; Array[f, 15, 0] (* Robert G. Wilson v, Jun 28 2012 *)
poly[q_] := 2^(q-n)/n*FunctionExpand[Sum[Binomial[n, k]*k^q, {k, 0, n}]]; T[q_, r_] := First[Take[CoefficientList[poly[q], n], {r+1, r+1}]]; Print[Table[Sum[T[2*x, k]*(-1)^(k+ x)*(2^k), {k, 0, 2*x-1}], {x, 1, 10}]]; (* John M. Campbell, Sep 15 2013 *)
a[ n_] := If[ n < 1, 0, With[ {k = 2 n + 1}, k! SeriesCoefficient[ Tan[x] / 2, {x, 0, k}]]] (* Michael Somos, Jan 21 2014 *)
a[ n_] := If[ n < 0, 0, With[ {k = 2 n}, k! SeriesCoefficient[ Tan[x]^2 / 2, {x, 0, k}]]] (* Michael Somos, Jan 21 2014 *)
a[0] = 0; a[n_] := (4^(n+1)-1)*Gamma[2*(n+1)]*Zeta[2*(n+1)]/Pi^(2*(n+1)); Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Feb 05 2016 *)

{a(n)=polcoeff( sum(m=1, n, x^m*prod(k=1, m, (2*k-1)^2/(1+(2*k-1)^2*x +x*O(x^n))) ), n)} \\ Paul D. Hanna, Feb 01 2013

G.f.: (1/2)*(tan(z))^2 = (z^2/(1-z^2)/2)*(1 +2*z^2/((z^2-1)*(G(0)-2*z^2)), G(k) = (k+2)*(2*k+3)-2*z^2+2*z^2*(k+2)*(2*k+3)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 15 2011
a(n) = (-1)^(n-1)*2^(2*n+1)*PolyLog(-2*n-1,-1) for n >= 1. - Peter Luschny, Jun 28 2012
O.g.f.: Sum_{n>=1} x^n * Product_{k=1..n} (2*k-1)^2 / (1 + (2*k-1)^2*x). - Paul D. Hanna, Feb 01 2013
G.f.: x/(Q(0)-x), where Q(k) = 1 + 2*x*(2*k+1)^2 - x*(2*k+3)^2*(1+x*(2*k+1)^2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 27 2013
a(n) ~ (2*n)! * n * 2^(2*n+3) / Pi^(2*n+2). - Vaclav Kotesovec, Aug 22 2014
a(n) = (4^(n+1)-1)*Gamma(2*(n+1))*zeta(2*(n+1))/Pi^(2*(n+1)) for n >= 1. - Jean-François Alcover, Feb 05 2016
From Peter Bala, Nov 16 2020: (Start)
a(n) = (1/2)*A000182(n+1) for n >= 1.
Conjectural o.g.f.: x/(1 + x - 9*x/(1 - 8*x/(1 + x - 25*x/(1 - 24*x/(1 + x - ... - (2*n+1)^2*x/(1 - 4*n*(n+1)*x/(1 + x - ... ))))))). (End)
a(n) = (-1)^(n-1)*PolyLog(-2*n - 1, i) for n >= 1. - Peter Luschny, Aug 12 2021

Extended and signs tested Mar 15 1997.

A111593 Triangle of tanh numbers.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, -2, 0, 1, 0, 0, -8, 0, 1, 0, 16, 0, -20, 0, 1, 0, 0, 136, 0, -40, 0, 1, 0, -272, 0, 616, 0, -70, 0, 1, 0, 0, -3968, 0, 2016, 0, -112, 0, 1, 0, 7936, 0, -28160, 0, 5376, 0, -168, 0, 1, 0, 0, 176896, 0, -135680, 0, 12432, 0, -240, 0, 1, 0, -353792, 0, 1805056, 0, -508640, 0, 25872

Offset: 0

Wolfdieter Lang, Aug 23 2005

Sheffer triangle associated to Sheffer triangle A060081.
For Sheffer triangles (matrices) see the explanation and S. Roman reference given under A048854.
In the umbral calculus (see the S. Roman reference) this triangle would be called associated for (1,arctanh(y)).
Without the n=0 row and m=0 column and unsigned, this is the Jabotinsky triangle A059419.
The inverse matrix of A with elements a(n,m), n,m>=0, is A111594.
The row polynomials p(n,x):=sum(a(n,m)*x^m,m=0..n), together with the row polynomials s(n,x) of A060081, satisfy the exponential (or binomial) convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), n>=0.
The row polynomials p(n,x) (defined above) have e.g.f. exp(x*tanh(y)).
Exponential Riordan array [1, tanh(x)], inverse of [1, arctanh(x)] which is A111594. - Paul Barry, May 30 2010
Also the Bell transform of A155585(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

			Binomial convolution of row polynomials: p(3,x)= -2*x+x^3; p(2,x)=x^2, p(1,x)= x, p(0,x)= 1, together with those from A060081:
s(3,x)= -5*x+x^3; s(2,x)= -1+x^2, s(1,x)= x, s(0,x)= 1;
therefore -5*(x+y)+(x+y)^3 = s(3,x+y) = 1*s(0,x)*p(3,y) + 3*s(1,x)*p(2,y) + 3*s(2,x)*p(1,y) +1*s(3,x)*p(0,y) = -2*y+y^3 + 3*x*y^2 + 3*(-1+x^2)*y + (-5*x+x^3).
From _Paul Barry_, May 30 2010: (Start)
Triangle begins:
  1;
  0,     1;
  0,     0,     1;
  0,    -2,     0,     1;
  0,     0,    -8,     0,     1;
  0,    16,     0,   -20,     0,     1;
  0,     0,   136,     0,   -40,     0,     1;
  0,  -272,     0,   616,     0,   -70,     0,     1;
  0,     0, -3968,     0,  2016,     0,  -112,     0,     1;
Production matrix begins:
  0,   1;
  0,   0,   1;
  0,  -2,   0,   1;
  0,   0,  -6,   0,   1;
  0,   0,   0, -12,   0,   1;
  0,   0,   0,   0, -20,   0,   1;
  0,   0,   0,   0,   0, -30,   0,   1;
  0,   0,   0,   0,   0,   0, -42,   0,   1;
  0,   0,   0,   0,   0,   0,   0, -56,   0,   1; (End)

W. Lang, First 10 rows.

Row sums: A003723. Unsigned row sums: A006229.
Cf. A059419, A060081, A111594.
Cf. A002378.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> 2^(n+1)*euler(n+1, 1), 9); # Peter Luschny, Jan 26 2016

t[0, 0] = 1; t[n_, m_] := Sum[ Binomial[k+m-1, m-1]*(k+m)!*(-1)^(k)*2^(n-k-m)*StirlingS2[n, k+m], {k, 0, n-m}]/m!; Table[t[n, m], {n, 0, 11}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 05 2013, after Vladimir Kruchinin *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[2^(#+1)*EulerE[#+1, 1]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

T(n,m):=if n=0 and m=0 then 1 else sum(binomial(k+m-1,m-1)*(k+m)!*(-1)^(k)*2^(n-k-m)*stirling2(n,k+m),k,0,n-m)/m!; /* Vladimir Kruchinin, Jun 09 2011 */

# uses[riordan_array from A256893]
riordan_array(1, tanh(x), 9, exp=true) # Peter Luschny, Apr 19 2015

E.g.f. for column m>=0: ((tanh(x))^m)/m!.
a(n, m) = coefficient of x^n of ((tanh(x))^m)/m!, n>=m>=0, else 0.
a(n, m) = a(n-1, m-1) - (m+1)*m*a(n-1, m+1), a(n, -1):=0, a(0, 0)=1, a(n, m)=0 for nT(n,m) = (Sum_{k=0..n-m} binomial(k+m-1,m-1)*(k+m)!*(-1)^k*2^(n-k-m)*stirling2(n,k+m))/m!, T(0,0)=1. - Vladimir Kruchinin, Jun 09 2011
With e.g.f. exp(x*tanh(t)) = sum(n>= 0, P(n,x)*t^n/n!), the lowering operator is L = arctanh(d/dx) = d/dx + (1/3)(d/dx)^3 + (1/5)(d/dx)^5 + ..., and the raising operator is R = x [1 - (d/dx)^2], where L P(n,x) = n P(n-1,x) and R P(n,x) = P(n+1,x), since the sequence is a binomial Sheffer sequence. - Tom Copeland, Oct 01 2015
The raising operator R = x - x D^2 in matrix form acting on an o.g.f. (formal power series) is the transpose of the production matrix M below. The linear term x is the diagonal of ones after transposition. The other transposed diagonal (A002378) comes from -x D^2 x^n = -n * (n-1) x^(n-1). Then P(n,x) = (1,x,x^2,..) M^n (1,0,0,..)^T. - Tom Copeland, Aug 17 2016

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A059420 A diagonal of A059419.

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A059421 A diagonal of A059419.

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A000182 Tangent (or "Zag") numbers: e.g.f. tan(x), also (up to signs) e.g.f. tanh(x).

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A007290 a(n) = 2*binomial(n,3).

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A155585 a(n) = 2^n*E(n, 1) where E(n, x) are the Euler polynomials.

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A006229 Expansion of e.g.f. exp( tan x ).

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