A099617 -id:A099617 - cp's OEIS frontend

A153641 Nonzero coefficients of the Swiss-Knife polynomials for the computation of Euler, tangent, and Bernoulli numbers (triangle read by rows).

1, 1, 1, -1, 1, -3, 1, -6, 5, 1, -10, 25, 1, -15, 75, -61, 1, -21, 175, -427, 1, -28, 350, -1708, 1385, 1, -36, 630, -5124, 12465, 1, -45, 1050, -12810, 62325, -50521, 1, -55, 1650, -28182, 228525, -555731, 1, -66, 2475, -56364, 685575, -3334386, 2702765, 1

Offset: 0

Peter Luschny, Dec 29 2008

In the following the expression [n odd] is 1 if n is odd, 0 otherwise.
(+) W_n(0) = E_n are the Euler (or secant) numbers A122045.
(+) W_n(1) = T_n are the signed tangent numbers, see A009006.
(+) W_{n-1}(1) n / (4^n - 2^n) = B_n gives for n > 1 the Bernoulli number A027641/A027642.
(+) W_n(-1) 2^{-n}(n+1) = G_n the Genocchi number A036968.
(+) W_n(1/2) 2^{n} are the signed generalized Euler (Springer) number, see A001586.
(+) | W_n([n odd]) | the number of alternating permutations A000111.
(+) | W_n([n odd]) / n! | for 0<=n the Euler zeta number A099612/A099617 (see Wikipedia on Bernoulli number). - Peter Luschny, Dec 29 2008
The diagonals in the full triangle (with zero coefficients) of the polynomials have the general form E(k)*binomial(n+k,k) (k>=0 fixed, n=0,1,...) where E(n) are the Euler numbers in the enumeration A122045. For k=2 we find the triangular numbers A000217 and for k=4 A154286. - Peter Luschny, Jan 06 2009
From Peter Bala, Jun 10 2009: (Start)
The Swiss-Knife polynomials W_n(x) may be expressed in terms of the Bernoulli polynomials B(n,x) as
... W_n(x) = 4^(n+1)/(2*n+2)*[B(n+1,(x+3)/4) - B(n+1,(x+1)/4)].
The Swiss-Knife polynomials are, apart from a multiplying factor, examples of generalized Bernoulli polynomials.
Let X be the Dirichlet character modulus 4 defined by X(4*n+1) = 1, X(4*n+3) = -1 and X(2*n) = 0. The generalized Bernoulli polynomials B(X;n,x), n = 1,2,..., associated with the character X are defined by means of the generating function
... t*exp(x*t)*(exp(t)-exp(3*t))/(exp(4*t)-1) = sum {n = 1..inf} B(X;n,x)*t^n/n!.
The first few values are B(X;1,x) = -1/2, B(X;2,x) = -x, B(X,3,x) = -3/2*(x^2-1) and B(X;4,x) = -2*(x^3-3*x).
In general, W_n(x) = -2/(n+1)*B(X;n+1,x).
For the theory of generalized Bernoulli polynomials associated to a periodic arithmetical function see [Cohen, Section 9.4].
The generalized Bernoulli polynomials may be used to evaluate twisted sums of k-th powers. For the present case the result is
sum{n = 0..4*N-1} X(n)*n^k = 1^k - 3^k + 5^k - 7^k + ... - (4*N-1)^k
= [B(X;k+1,4*N) - B(X;k+1,0)]/(k+1) = [W_k(0) - W_k(4*N)]/2.
For the proof apply [Cohen, Corollary 9.4.17 with m = 4 and x = 0].
The generalized Bernoulli polynomials and the Swiss-Knife polynomials are also related to infinite sums of powers through their Fourier series - see the formula section below. For a table of the coefficients of generalized Bernoulli polynomials attached to a Dirichlet character modulus 8 see A151751.
(End)
The Swiss-Knife polynomials provide a general formula for alternating sums of powers similar to the formula which are provided by the Bernoulli polynomials for non-alternating sums of powers (see the Luschny link). Sequences covered by this formula include A001057, A062393, A062392, A011934, A144129, A077221, A137501, A046092. - Peter Luschny, Jul 12 2009
The greatest common divisor of the nonzero coefficients of the decapitated Swiss-Knife polynomials is exp(Lambda(n)), where Lambda(n) is the von Mangoldt function for odd primes, symbolically:
gcd(coeffs(SKP_{n}(x) - x^n)) = A155457(n) (n>1). - Peter Luschny, Dec 16 2009
Another version is at A119879. - Philippe Deléham, Oct 26 2013

			1
x
x^2  -1
x^3  -3x
x^4  -6x^2   +5
x^5 -10x^3  +25x
x^6 -15x^4  +75x^2  -61
x^7 -21x^5 +175x^3 -427x

H. Cohen, Number Theory - Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics. Springer-Verlag. [From Peter Bala, Jun 10 2009]

G. C. Greubel, Table of n, a(n) for the first 76 rows, flattened
Ayse Yilmaz Ceylan and Yilmaz Simsek, Formulae for Generalization of Touchard Polynomials with Their Generating Functions, Symmetry (2025) Vol. 17, Issue 7, Art. No. 1126. See Eq. 28 and after.
Kwang-Wu Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.
Suyoung Choi and Hanchul Park, A new graph invariant arises in toric topology, arXiv preprint arXiv:1210.3776 [math.AT], 2012.
Leonhard Euler (1735), De summis serierum reciprocarum, Opera Omnia I.14, E 41, 73-86; On the sums of series of reciprocals, arXiv:math/0506415 [math.HO], 2005-2008.
A. Hodges and C. V. Sukumar, Bernoulli, Euler, permutations and quantum algebras, Proc. R. Soc. A Oct. 2007 vol 463 no. 463 2086 2401-2414. [Added by Tom Copeland, Aug 31 2015]
Peter Luschny, The Swiss-Knife polynomials.
Peter Luschny, Swiss-Knife polynomials and Euler numbers
Wikipedia, Bernoulli number
J. Worpitzky, Studien über die Bernoullischen und Eulerschen Zahlen, Journal für die reine und angewandte Mathematik, 94 (1883), 203-232.

Cf. A151751, A162590, A000111, A001586, A009006, A027641/A027642, A036968, A099612/A099617, A119879, A104033.
W_n(k), k=0,1,...
W_0:  1,  1,  1,  1,   1,   1, ........ A000012
W_1:  0,  1,  2,  3,   4,   5, ........ A001477
W_2: -1,  0,  3,  8,  15,  24, ........ A067998
W_3:  0, -2,  2, 18,  52, 110, ........ A121670
W_4:  5,  0, -3, 32, 165, 480, ........
W_n(k), n=0,1,...
k=0:  1,  0, -1,  0,   5,   0, -61, ... A122045
k=1:  1,  1,  0, -2,   0,  16,   0, ... A155585
k=2:  1,  2,  3,  2,  -3,   2,  63, ... A119880
k=3:  1,  3,  8, 18,  32,  48, 128, ... A119881
k=4:  1,  4, 15, 52, 165, 484, ........         [Peter Luschny, Jul 07 2009]

w := proc(n,x) local v,k,pow,chen; pow := (a,b) -> if a = 0 and b = 0 then 1 else a^b fi; chen := proc(m) if irem(m+1,4) = 0 then RETURN(0) fi; 1/((-1)^iquo(m+1,4) *2^iquo(m,2)) end; add(add((-1)^v*binomial(k,v)*pow(v+x+1,n)*chen(k),v=0..k), k=0..n) end:
# Coefficients with zeros:
seq(print(seq(coeff(i!*coeff(series(exp(x*t)*sech(t),t,16),t,i),x,i-n),n=0..i)), i=0..8);
# Recursion
W := proc(n,z) option remember; local k,p;
if n = 0 then 1 else p := irem(n+1,2);
z^n - p + add(`if`(irem(k,2)=1,0,
W(k,0)*binomial(n,k)*(power(z,n-k)-p)),k=2..n-1) fi end:
# Peter Luschny, edited and additions Jul 07 2009, May 13 2010, Oct 24 2011

max = 9; rows = (Reverse[ CoefficientList[ #, x]] & ) /@ CoefficientList[ Series[ Exp[x*t]*Sech[t], {t, 0, max}], t]*Range[0, max]!; par[coefs_] := (p = Partition[ coefs, 2][[All, 1]]; If[ EvenQ[ Length[ coefs]], p, Append[ p, Last[ coefs]]]); Flatten[ par /@ rows] (* Jean-François Alcover, Oct 03 2011, after g.f. *)
sk[n_, x_] := Sum[Binomial[n, k]*EulerE[k]*x^(n-k), {k, 0, n}]; Table[CoefficientList[sk[n, x], x] // Reverse // Select[#, # =!= 0 &] &, {n, 0, 13}] // Flatten (* Jean-François Alcover, May 21 2013 *)
Flatten@Table[Binomial[n, 2k] EulerE[2k], {n, 0, 12}, {k, 0, n/2}](* Oliver Seipel, Jan 14 2025 *)

def A046978(k):
    if k % 4 == 0:
        return 0
    return (-1)**(k // 4)
def A153641_poly(n, x):
    return expand(add(2**(-(k // 2))*A046978(k+1)*add((-1)**v*binomial(k,v)*(v+x+1)**n for v in (0..k)) for k in (0..n)))
for n in (0..7): print(A153641_poly(n, x))  # Peter Luschny, Oct 24 2011

W_n(x) = Sum_{k=0..n}{v=0..k} (-1)^v binomial(k,v)*c_k*(x+v+1)^n where c_k = frac((-1)^(floor(k/4))/2^(floor(k/2))) [4 not div k] (Iverson notation).
From Peter Bala, Jun 10 2009: (Start)
E.g.f.: 2*exp(x*t)*(exp(t)-exp(3*t))/(1-exp(4*t))= 1 + x*t + (x^2-1)*t^2/2! + (x^3-3*x)*t^3/3! + ....
W_n(x) = 1/(2*n+2)*Sum_{k=0..n+1} 1/(k+1)*Sum_{i=0..k} (-1)^i*binomial(k,i)*((x+4*i+3)^(n+1) - (x+4*i+1)^(n+1)).
Fourier series expansion for the generalized Bernoulli polynomials:
B(X;2*n,x) = (-1)^n*(2/Pi)^(2*n)*(2*n)! * {sin(Pi*x/2)/1^(2*n) - sin(3*Pi*x/2)/3^(2*n) + sin(5*Pi*x/2)/5^(2*n) - ...}, valid for 0 <= x <= 1 when n >= 1.
B(X;2*n+1,x) = (-1)^(n+1)*(2/Pi)^(2*n+1)*(2*n+1)! * {cos(Pi*x/2)/1^(2*n+1) - cos(3*Pi*x/2)/3^(2*n+1) + cos(5*Pi*x/2)/5^(2*n+1) - ...}, valid for 0 <= x <= 1 when n >= 1 and for 0 <= x < 1 when n = 0.
(End)
E.g.f.: exp(x*t) * sech(t). - Peter Luschny, Jul 07 2009
O.g.f. as a J-fraction: z/(1-x*z+z^2/(1-x*z+4*z^2/(1-x*z+9*z^2/(1-x*z+...)))) = z + x*z^2 + (x^2-1)*z^3 + (x^3-3*x)*z^4 + .... - Peter Bala, Mar 11 2012
Conjectural o.g.f.: Sum_{n >= 0} (1/2^((n-1)/2))*cos((n+1)*Pi/4)*( Sum_{k = 0..n} (-1)^k*binomial(n,k)/(1 - (k + x)*t) ) = 1 + x*t + (x^2 - 1)*t^2 + (x^3 - 3*x)*t^3 + ... (checked up to O(t^13)), which leads to W_n(x) = Sum_{k = 0..n} 1/2^((k - 1)/2)*cos((k + 1)*Pi/4)*( Sum_{j = 0..k} (-1)^j*binomial(k, j)*(j + x)^n ). - Peter Bala, Oct 03 2016

A099612 Numerators of the coefficients in the Taylor expansion of sec(x) + tan(x) around x=0.

Original entry on oeis.org

1, 1, 1, 1, 5, 2, 61, 17, 277, 62, 50521, 1382, 540553, 21844, 199360981, 929569, 3878302429, 6404582, 2404879675441, 443861162, 14814847529501, 18888466084, 69348874393137901, 113927491862, 238685140977801337, 58870668456604, 4087072509293123892361

Offset: 0

N. J. A. Sloane, Nov 18 2004

			1 + x + 1/2*x^2 + 1/3*x^3 + 5/24*x^4 + 2/15*x^5 + 61/720*x^6 + 17/315*x^7 + ...
1, 1, 1/2, 1/3, 5/24, 2/15, 61/720, 17/315, 277/8064, 62/2835, 50521/3628800, 1382/155925, 540553/95800320, ... = A099612/A099617

Seiichi Manyama, Table of n, a(n) for n = 0..487
L. Euler, On the sums of series of reciprocals, par. 13, arXiv:math/0506415 [math.HO], 2005-2008.
L. Euler, De summis serierum reciprocarum, E41, par. 13, Euler Archive.
Peter Luschny, The Euler-Bernoulli diamond and the lost Bernoulli numbers.

Cf. A000111, A099617, A279107.

R := n -> (cos(n*Pi/2)+sin(n*Pi/2))*(4^(n+1)-2^(n+1))*((Zeta(0,-n,3/4)-Zeta(0,-n,1/4))/(2^(-n)-2)-Zeta(-n))/GAMMA(n+1):
[1, seq(numer(R(n)), n=1..19)]; # Peter Luschny, Aug 25 2015
# From Peter Luschny, Aug 03 2017: (Start) By recurrence:
S := proc(n, k) option remember; if k = 0 then `if`(n = 0, 1, 0) else
S(n, k - 1) + S(n - 1, n - k) fi end: A099612 := n -> numer(S(n, n)/n!):
seq(A099612(n), n=0..26);
# or evaluating polynomials at -i:
P := proc(n, x) local k, j; add(add((-1)^j*2^(-k)*binomial(k,j)*(k-2*j)^n*
x^(n-k), j=0..k), k=0..n) end: R := n -> `if`(n = 0, 1, P(n-1, -I)/ n!):
seq(numer(R(n)), n=0..26);
# or with the Euler polynomials:
ep := n -> `if`(n=0,1,2^n*(euler(n,1/2)-euler(n,1))*(-1)^iquo(n+1,2)):
a := n -> numer(ep(n)/n!): seq(a(n), n=0..26); # (End)

nn = 26; Numerator[CoefficientList[Series[Sec[x] + Tan[x], {x, 0, nn}], x]] (* T. D. Noe, Jul 24 2013 *)
Table[If[n==0,1,2 I^(n+1) PolyLog[-n, -I] / n!], {n,0,26}] // Numerator (* Peter Luschny, Aug 03 2017 *)
Table[(1 + Mod[n,2])LerchPhi[(-1)^(n+1), n+1, 1/2]/Pi^(n+1), {n, 0, 26}] // Numerator (* Peter Luschny, Aug 03 2017 *)

Let R(x) = (-1)^floor(x/2)*(4^(x+1)-2^(x+1))*((HurwitzZeta(-x,3/4) - HurwitzZeta(-x,1/4)) /(2^(-x)-2)-Zeta(-x))/Gamma(x+1) then a(n) = numerator(R(n)) and A099617(n) = denominator(R(n)) for n>=1. - Peter Luschny, Aug 25 2015
Let F(x,t) = exp(-I*t*x)*(1+(exp(exp(I*t))-1)/(exp(2*exp(I*t))+1)) and r(x) = ((cos(x*Pi/2)+sin(x*Pi/2))/Pi)*Integral_{t=0..2*Pi} F(x,t) then a(n) = numerator(r(n)) and A099617(n) = denominator(r(n)) for n>=1. - Peter Luschny, Aug 25 2015
a(n)/A099617(n) = A000111(n)/n!. - Seiichi Manyama, Jan 27 2017
From Peter Luschny, Aug 03 2017: (Start)
a(n) = numerator(2*i^(n+1)*PolyLog(-n, -i)/n!) for n>0.
a(n) = numerator(2^n*|Euler(n,1/2) - Euler(n,1)|/n!) for n>0 where Euler(n,x) are the Euler polynomials. (End)
Conjecture: For n >= 0, (-1)^n * a(n+1) is the numerator of the n-th term of the Taylor expansion of 1/(1 + sin(x)) around x = 0. [This is based on the fact that (sec(x) + tan(x))' = 1/(1 + sin(-x)). Clark Kimberling in A279107 states my conjecture as a fact, but no proof or reference is given.] - Petros Hadjicostas, Oct 06 2019

A328186 Write 1/(1 + sin x) = Product_{n>=1} (1 + f_n x^n); a(n) = denominator(f_n).

Original entry on oeis.org

1, 1, 6, 6, 120, 360, 5040, 2520, 72576, 1814400, 39916800, 59875200, 1245404160, 21794572800, 1307674368000, 81729648000, 71137485619200, 3201186852864000, 121645100408832000, 12164510040883200, 10218188434341888000, 281000181944401920000, 25852016738884976640000

Offset: 1

Petros Hadjicostas, Oct 06 2019

The recurrence about (A(m,n): m,n >= 1) in the Formula section follows from Theorem 3 in Gingold et al. (1988); see also Gingold and Knopfmacher (1995, p. 1222). A(m=1,n) equals the n-th coefficient of the Taylor expansion of 1/(1 + sin(x)). For that coefficient, we use a modification of a formula by Peter Luschny in the documentation of sequences A099612 and A099617.
Write 1 + sin x = Product_{n>=1} (1 + g_n * x^n). We have A170914(n) = numerator(g_n) and A170915(n) = denominator(g_n).
Gingold and Knopfmacher (1995) and Alkauskas (2008, 2009) proved that f_n = -g_n for n odd, and Sum_{s|n} (-g_{n/s})^s/s = -Sum_{s|n} (-f_{n/s})^s/s. [We caution that different authors may use -g_n for g_n, or -f_n for f_n, or both.]
Wolfdieter Lang (see the link below) examined inverse power product expansions both for ordinary g.f.'s and for exponential g.f.'s. He connects inverse power product expansions to unital series associated to (infinite dimensional) Witt vectors and to the so-called "Somos transformation".
There are more formulas for f_n and g_n in the references listed below. In all cases, we assume the g.f.'s are unital, i.e., the g.f.'s start with a constant 1.

			f_n = -1, 1, 1/6, 5/6, 19/120, -47/360, 659/5040, 1837/2520, 7675/72576, -154729/1814400, 3578279/39916800, 3984853/59875200, 95259767/1245404160, ...

Giedrius Alkauskas, One curious proof of Fermat's little theorem, arXiv:0801.0805 [math.NT], 2008.
Giedrius Alkauskas, A curious proof of Fermat's little theorem, Amer. Math. Monthly 116(4) (2009), 362-364.
H. Gingold, H. W. Gould, and Michael E. Mays, Power Product Expansions, Utilitas Mathematica 34 (1988), 143-161.
H. Gingold and A. Knopfmacher, Analytic properties of power product expansions, Canad. J. Math. 47 (1995), 1219-1239.
W. Lang, Recurrences for the general problem, 2009.

Numerators are in A328191.

Cf. A099612, A099617, A170914, A170915, A279107.

# Calculates the fractions f_n (choose L much larger than M):
PPE := proc(L, M)
local t1, t0, g, t2, n, t3;
if L < 2.5*M then print("Choose larger value for L");
else
t1 := 1/(1 + sin(x));
t0 := series(t1, x, L);
f := []; t2 := t0;
for n to M do
t3 := coeff(t2, x, n);
t2 := series(t2/(1 + t3*x^n), x, L);
f := [op(f), t3];
end do;
end if;
[seq(f[n], n = 1 .. nops(f))];
end proc;
# Calculates the denominators of f_n:
h := map(denom, PPE(100, 40)); # Petros Hadjicostas, Oct 06 2019 by modifying N. J. A. Sloane's program from A170912 and A170913.

A[m_, n_] :=
  A[m, n] =
   Which[m == 1, 2*(-1)^n*I^(n + 2)*PolyLog[-(n + 1), -I]/n!,
    m > n >= 1, 0, True,
    A[m - 1, n] - A[m - 1, m - 1]*A[m, n - m + 1]];
a[n_] := Denominator[A[n, n]];
a /@ Range[1, 55] (* Petros Hadjicostas, Oct 06 2019 using a program by Jean-François Alcover and a formula from A099612 and A099617 *)

a(2*n + 1) = A170915(2*n + 1) for n >= 0.
Define (A(m,n): n,m >= 1) by A(m=1, n) = 2 * (-1)^n * i^(n + 2) * PolyLog(-(n + 1), -i)/n! for n >= 1 (with i := sqrt(-1)), A(m,n) = 0 for m > n >= 1 (upper triangular), and A(m,n) = A(m-1,n) - A(m-1,m-1) * A(m,n-m+1) for n >= m >= 2. Then f_n = A(n,n) and thus a(n) = denominator(A(n,n)).
If we write 1 + sin x = Product_{n>=1} (1 + g_n * x^n) and we know (g_n: n >= 1), then f_n = -g_n + Sum_{s|n, s > 1} (1/s) * ((-f_{n/s})^s + (-g_{n/s})^s). This proves of course that f_n = -g_n for n odd.

A328191 Write 1/(1 + sin x) = Product_{n>=1} (1 + f_n x^n); a(n) = numerator(f_n).

Original entry on oeis.org

-1, 1, 1, 5, 19, -47, 659, 1837, 7675, -154729, 3578279, 3984853, 95259767, -1364856587, 85215100151, 46900690817, 4180679675171, -157452879103733, 6399968826052559, 893237028559439, 478887035449041839, -11642446645024341437, 1123931378903214542099, 15392026390952264029

Offset: 1

Petros Hadjicostas, Oct 07 2019

The recurrence about (A(m,n): m,n >= 1) in the Formula section follows from Theorem 3 in Gingold et al. (1988); see also Gingold and Knopfmacher (1995, p. 1222). A(m=1,n) equals the n-th coefficient of the Taylor expansion of 1/(1 + sin(x)). For that coefficient, we use a modification of a formula by Peter Luschny in the documentation of sequences A099612 and A099617.
Write 1 + sin x = Product_{n>=1} (1 + g_n * x^n). We have A170914(n) = numerator(g_n) and A170915(n) = denominator(g_n).
Gingold and Knopfmacher (1995) and Alkauskas (2008, 2009) proved that f_n = -g_n for n odd, and Sum_{s|n} (-g_{n/s})^s/s = -Sum_{s|n} (-f_{n/s})^s/s. [We caution that different authors may use -g_n for g_n, or -f_n for f_n, or both.]
Wolfdieter Lang (see the link below) examined inverse power product expansions both for ordinary g.f.'s and for exponential g.f.'s. He connects inverse power product expansions to unital series associated to (infinite dimensional) Witt vectors and to the so-called "Somos transformation".
There are more formulas for f_n and g_n in the references listed below. In all cases, we assume the g.f.'s are unital, i.e., the g.f.'s start with a constant 1.

			f_n = -1, 1, 1/6, 5/6, 19/120, -47/360, 659/5040, 1837/2520, 7675/72576, -154729/1814400, 3578279/39916800, 3984853/59875200, 95259767/1245404160, ...

Giedrius Alkauskas, One curious proof of Fermat's little theorem, arXiv:0801.0805 [math.NT], 2008.
Giedrius Alkauskas, A curious proof of Fermat's little theorem, Amer. Math. Monthly 116(4) (2009), 362-364.
H. Gingold, H. W. Gould, and Michael E. Mays, Power Product Expansions, Utilitas Mathematica 34 (1988), 143-161.
H. Gingold and A. Knopfmacher, Analytic properties of power product expansions, Canad. J. Math. 47 (1995), 1219-1239.
W. Lang, Recurrences for the general problem, 2009.

Denominators are in A328186.

Cf. A099612, A099617, A170914, A170915, A279107.

# Calculates the fractions f_n (choose L much larger than M):
PPE := proc(L, M)
local t1, t0, g, t2, n, t3;
if L < 2.5*M then print("Choose larger value for L");
else
t1 := 1/(1 + sin(x));
t0 := series(t1, x, L);
f := []; t2 := t0;
for n to M do
t3 := coeff(t2, x, n);
t2 := series(t2/(1 + t3*x^n), x, L);
f := [op(f), t3];
end do;
end if;
[seq(f[n], n = 1 .. nops(f))];
end proc;
# Calculates the numerators of f_n:
h:=map(numer, PPE(100, 40)); # Petros Hadjicostas, Oct 07 2019 by modifying N. J. A. Sloane's program from A170912 and A170913.

A[m_, n_] :=
  A[m, n] =
   Which[m == 1, 2*(-1)^n*I^(n + 2)*PolyLog[-(n + 1), -I]/n!,
    m > n >= 1, 0, True,
    A[m - 1, n] - A[m - 1, m - 1]*A[m, n - m + 1]];
a[n_] := Numerator[A[n, n]];
a /@ Range[1, 55] (* Petros Hadjicostas, Oct 07 2019 using a program by Jean-François Alcover and a formula from A099612 and A099617 *)

a(2*n + 1) = -A170914(2*n + 1) for n >= 0.
Define (A(m,n): n,m >= 1) by A(m=1, n) = 2 * (-1)^n * i^(n + 2) * PolyLog(-(n + 1), -i)/n! for n >= 1 (with i := sqrt(-1)), A(m,n) = 0 for m > n >= 1 (upper triangular), and A(m,n) = A(m-1,n) - A(m-1,m-1) * A(m,n-m+1) for n >= m >= 2. Then f_n = A(n,n) and thus a(n) = numerator(A(n,n)).
If we write 1 + sin x = Product_{n>=1} (1 + g_n * x^n) and we know (g_n: n >= 1), then f_n = -g_n + Sum_{s|n, s > 1} (1/s) * ((-f_{n/s})^s + (-g_{n/s})^s). This proves of course that f_n = -g_n for n odd.

cp's OEIS Frontend

A153641 Nonzero coefficients of the Swiss-Knife polynomials for the computation of Euler, tangent, and Bernoulli numbers (triangle read by rows).

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A099612 Numerators of the coefficients in the Taylor expansion of sec(x) + tan(x) around x=0.

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A328186 Write 1/(1 + sin x) = Product_{n>=1} (1 + f_n x^n); a(n) = denominator(f_n).

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A328191 Write 1/(1 + sin x) = Product_{n>=1} (1 + f_n x^n); a(n) = numerator(f_n).

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A172170 1 followed by the duplicated entries of A090368.

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