A002436 -id:A002436 - cp's OEIS frontend

A000819 E.g.f.: cos(x)^2 / cos(2x) = Sum_{n >= 0} a(n) * x^(2n) / (2n)!.

1, 2, 40, 1952, 177280, 25866752, 5535262720, 1633165156352, 635421069967360, 315212388819402752, 194181169538675507200, 145435130631317935357952, 130145345400688287667978240, 137139396592145493713802493952

Offset: 0

N. J. A. Sloane

nonn

			G.f. = 1 + 2*x + 40*x^2 + 1952*x^3 + 177280*x^4 + 25866752*x^5 + ... - _Michael Somos_, Apr 04 2017

R. J. Mathar, Table of n, a(n) for n = 0..200

Essentially the same as A000816.
Second column of array A103905.
Cf. A000364, A002436.

With[{nn=30},Take[CoefficientList[Series[Cos[x]^2/Cos[2x],{x,0,nn}],x] Range[ 0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Jul 06 2014 *)
a[ n_] := If[ n < 0, 0, With[ {m = 2 n}, m! SeriesCoefficient[ 1 / (1 - Tan[x]^2), {x, 0, m}]]]; (* Michael Somos, Apr 04 2017 *)

{a(n) = my(m); if( n<0, 0, m = 2*n; m! * polcoeff( 1 / (1 - tan(x + x * O(x^m))^2), m))}; /* Michael Somos, Apr 04 2017 */

E.g.f.: cos(x)^2/cos(2x)=1/Q(0)+1/2;   Q(k)=1+1/(1-2*(x^2)/(2*(x^2)-(k+1)*(2k+1)/Q(k+1)));  (continued fraction). - Sergei N. Gladkovskii, Nov 18 2011
From Michael Somos, Apr 04 2017: (Start)
E.g.f.: cos(x)^2 / cos(2*x) = (1 + sec(2*x)) / 2 = tan(2*x) / (2 * tan(x)) = 1 / (1 - tan(x)^2).
a(n) = A000816(n) unless n=0.
a(n) = 1/2 * A002436(n) unless n=0.
a(n) = 2^(2*n - 1) * A000364(n). (End)

A117437 Expansion of e.g.f.: exp(x)sec(2x).

Original entry on oeis.org

1, 1, 5, 13, 105, 441, 5165, 30213, 469585, 3529201, 68525525, 629401213, 14664091065, 159175688361, 4326609913085, 54189700721013, 1683369010256545, 23894940183997921, 835066388382183845, 13248060325188261613

Offset: 0

Paul Barry, Mar 16 2006

Row sums of A117436.

Binomial transform of A002436 (with interpolated zeros).

G. C. Greubel, Table of n, a(n) for n = 0..425

Cf. A002436, A117436.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(x)*Sec(2*x) ))); // G. C. Greubel, May 31 2021

With[{nn=30},CoefficientList[Series[Exp[x]Sec[2x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 13 2011 *)

my(x='x+O('x^30)); Vec(serlaplace(exp(x)/cos(2*x))) \\ Michel Marcus, Jun 01 2021

[factorial(n)*( exp(x)*sec(2*x) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, May 31 2021

a(n) ~ n! * 2^(2*n+1) * (exp(Pi/4) + (-1)^n*exp(-Pi/4)) / Pi^(n+1). - Vaclav Kotesovec, Aug 04 2014

A291260 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 - 2^kx/(1 - 4^kx/(1 - 6^kx/(1 - 8^kx/(1 - 10^k*x/(1 - ...)))))).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 4, 12, 5, 1, 8, 80, 120, 14, 1, 16, 576, 3904, 1680, 42, 1, 32, 4352, 152064, 354560, 30240, 132, 1, 64, 33792, 6492160, 99422208, 51733504, 665280, 429, 1, 128, 266240, 290488320, 31832735744, 130292416512, 11070525440, 17297280, 1430

Offset: 0

Ilya Gutkovskiy, Aug 21 2017

			Square array begins:
:  1,     1,        1,            1,               1, ...
:  1,     2,        4,            8,              16, ...
:  2,    12,       80,          576,            4352, ...
:  5,   120,     3904,       152064,         6492160, ...
: 14,  1680,   354560,     99422208,     31832735744, ...
: 42, 30240, 51733504, 130292416512, 390365719822336, ...

Columns k=0-2 give A000108, A001813, A002436.
Main diagonal gives A291331.
Cf. A000079 (row 1), A063481 (row 2), A290569, A291261.

Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[-(2 i)^k x, 1, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 8}, {n, 0, j}] // Flatten

G.f. of column k: 1/(1 - 2^k*x/(1 - 4^k*x/(1 - 6^k*x/(1 - 8^k*x/(1 - 10^k*x/(1 - ...)))))), a continued fraction.

A138076 Triangle read by rows: T(n, k) = (-1)^(n+k) * A060187(n+1, k+1).

Original entry on oeis.org

1, -1, 1, 1, -6, 1, -1, 23, -23, 1, 1, -76, 230, -76, 1, -1, 237, -1682, 1682, -237, 1, 1, -722, 10543, -23548, 10543, -722, 1, -1, 2179, -60657, 259723, -259723, 60657, -2179, 1, 1, -6552, 331612, -2485288, 4675014, -2485288, 331612, -6552, 1, -1, 19673, -1756340, 21707972, -69413294, 69413294, -21707972, 1756340, -19673, 1

Offset: 0

Roger L. Bagula, Nov 26 2009

Former name: A signed version of A060187 obtained by taking the Z-transform of p(t,x) = exp(t*(1+2*x)). - G. C. Greubel, Jul 21 2024

			Triangle begins as:
   1;
  -1,     1;
   1,    -6,      1;
  -1,    23,    -23,        1;
   1,   -76,    230,      -76,       1;
  -1,   237,  -1682,     1682,    -237,        1;
   1,  -722,  10543,   -23548,   10543,     -722,      1;
  -1,  2179, -60657,   259723, -259723,    60657,  -2179,     1;
   1, -6552, 331612, -2485288, 4675014, -2485288, 331612, -6552, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000165, A002436, A060187, A177043, A178118.

A060187:= func< n,k | (&+[(-1)^(k-j)*Binomial(n, k-j)*(2*j-1)^(n-1): j in [1..k]]) >;
A138076:= func< n,k | (-1)^(n+k)*A060187(n+1,k+1) >;
[A138076(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 21 2024

p[t_] = Exp[t]*x/(Exp[2*t] + x);
Table[CoefficientList[(n!*(1+x)^(n+1)/x)*SeriesCoefficient[Series[p[ t], {t,0,30}], n], x], {n,0,12}]//Flatten

@CachedFunction
def t(n,k): # t = A060187
    if k==1 or k==n: return 1
    return (2*(n-k)+1)*t(n-1, k-1) + (2*k-1)*t(n-1, k)
def A138076(n,k): return (-1)^(n+k)*t(n+1,k+1)
flatten([[A138076(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 21 2024

T(n, k) = (-1)^(n+k) * A060187(n+1, k+1).
From G. C. Greubel, Jul 21 2024: (Start)
T(2*n, n) = (-1)^n * A177043(n).
Sum_{k=0..n} T(n, k) = (1/2)*(1 + (-1)^n)*(-1)^floor((n+ 1)/2) * A002436(floor(n/2)).
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n * A000165(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n * A178118(n+1). (End)

A335257 Numerators of expansion of arctanh(tan(x)) (odd powers only).

Original entry on oeis.org

1, 2, 2, 244, 554, 202084, 166324, 1594887848, 456270874, 9619518701764, 59259390118004, 554790995145103208, 954740563911205348, 32696580074344991138888, 3636325637469705598456, 7064702291984369672858925136, 4176926860695042104392112698

Offset: 1

Denis Roegel, May 28 2020

nonn

The numerators of a series used by Johann Heinrich Lambert (1728-1777) in expressing the relationship between a circular sector and a hyperbolic sector.

			arctan(tanh(x)) = x - 2/3*x^3 + 2/3*x^5 - 244/315*x^7 + 554/567*x^9 ...
arctanh(tan(x)) = x + 2/3*x^3 + 2/3*x^5 + 244/315*x^7 + 554/567*x^9 ...

Johann Heinrich Lambert, Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques, Histoire de l'Académie Royale des Sciences et Belles-Lettres, 1761, volume XVII, Berlin, 1768, pp. 265-322. See also.
Denis Roegel, Lambert's proof of the irrationality of Pi: Context and translation, hal-02984214 [math.HO], 2020.

Cf. A002436, A335258 (denominators).

Numerator @ CoefficientList[Series[ArcTanh[Tan[x]], {x, 0, 34}], x][[2 ;; -1 ;; 2]] (* Amiram Eldar, May 30 2020 *)

a(n)={numerator((-1)^(n-1)*(polcoef(atan(tanh(x + O(x^(2*n)))), 2*n-1)))} \\ Andrew Howroyd, May 29 2020

a(n)/A335258(n) = A002436(n-1)/(2*n-1)!. - Andrew Howroyd, May 29 2020

Terms a(9) and beyond from Andrew Howroyd, May 29 2020

A371688 Triangle read by rows: T(n, k) = (2n + 1)! [y^(2k)] [x^(2n+1)] arctan(sec(xy)sinh(x)).

Original entry on oeis.org

1, -1, 3, 5, -50, 25, -61, 1281, -2135, 427, 1385, -49860, 174510, -116340, 12465, -50521, 2778655, -16671930, 23340702, -8335965, 555731, 2702765, -210815670, 1932476975, -4637944740, 3478458555, -772990790, 35135945

Offset: 0

Peter Luschny, Apr 03 2024

Expansion of the exponential generating function arctan(sec(x*y)*sinh(x)), nonzero terms only.

			Triangle starts:
  [0]      1;
  [1]     -1,       3;
  [2]      5,     -50,        25;
  [3]    -61,    1281,     -2135,      427;
  [4]   1385,  -49860,    174510,  -116340,    12465;
  [5] -50521, 2778655, -16671930, 23340702, -8335965, 555731;

Cf. A000364 (column 0), A009843 (main diagonal), A012816 (row sums), A002436 (alternating row sums).

egf := arctan(sec(x*y)*sinh(x)):
serx := simplify(series(egf, x, 26)): coeffx := n -> n!*coeff(serx, x, n):
seq(lprint(seq(coeff(coeffx(2*n + 1), y, 2*k), k = 0..n)), n = 0..7);

T[n_,k_]:=(-1)^k*Binomial[2*n+1,2*k]*EulerE[2*n];Flatten[Table[T[n,k],{n,0,6},{k,0,n}]] (* Detlef Meya, Apr 07 2024 *)

T(n, k) = (-1)^k*binomial(2*n + 1, 2*k)*Euler(2*n). - Detlef Meya, Apr 07 2024

A256679 A signed triangle of V. I. Arnold for the Springer numbers (A001586).

Original entry on oeis.org

1, 1, 0, -2, -3, -3, -8, -6, -3, 0, 40, 48, 54, 57, 57, 256, 216, 168, 114, 57, 0, -1952, -2208, -2424, -2592, -2706, -2763, -2763, -17408, -15456, -13248, -10824, -8232, -5526, -2763, 0, 177280, 194688, 210144, 223392, 234216, 242448, 247974, 250737, 250737

Offset: 0

Vladimir Kruchinin, Apr 07 2015

This triangle is denoted R(b) on page 6 of the Arnold reference.
Unsigned version of triangle is triangle of A202818.
First column (m=0) is A000828. - Robert Israel, Apr 08 2015

			Triangle begins:
    1;
    1,   0;
   -2,  -3,  -3;
   -8,  -6,  -3,   0;
   40,  48,  54,  57, 57;
  256, 216, 168, 114, 57, 0;

V. I. Arnold, The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups (in Russian), Uspekhi Mat. nauk., 47 (#1, 1992), 3-45 = Russian Math. Surveys, Vol. 47 (1992), 1-51. (See page 6)

Cf. A000828, A001586, A002436, A008280, A122045, A202818, A256665.

T:= (n,m) -> add(add(4^i*euler(2*i)*binomial(2*k,2*i)*binomial(n-m,2*k-m),i=0..k),k=floor(m/2)..floor(n/2)):
seq(seq(T(n,m),m=0..n),n=0..10); # Robert Israel, Apr 08 2015
# Second program, about 100 times faster than the first for the first 100 rows.
Triangle := proc(len) local s, A, n, k;
A := Array(0..len-1,[1]); lprint(A[0]);
for n from 1 to len-1 do
   if n mod 2 = 1 then s := 0 else
   s := 2^(3*n+1)*(Zeta(0,-n,1/8)-Zeta(0,-n,5/8)) fi;
   A[n] := s;
   for k from n-1 by -1 to 0 do
       s := s + A[k]; A[k] := s od;
   lprint(seq(A[k], k=0..n));
od end:
Triangle(100); # Peter Luschny, Apr 08 2015

T[n_, m_] := Sum[4^i EulerE[2i] Binomial[2k, 2i] Binomial[n-m, 2k-m], {k, Floor[m/2], n/2}, {i, 0, k}];
Table[T[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 12 2019 *)

T(n,m):=(sum((sum(4^i*euler(2*i)*binomial(2*k,2*i),i,0,k))*binomial(n-m,2*k-m),k,floor(m/2),n/2));

def triangle(len):
    L = [1]; print(L)
    for n in range(1,len):
        if is_even(n):
            s = 2^(3*n+1)*(hurwitz_zeta(-n,1/8)-hurwitz_zeta(-n,5/8))
        else: s = 0
        L.append(s)
        for k in range(n-1,-1,-1):
            s = s + L[k]; L[k] = s
        print(L)
triangle(7) # Peter Luschny, Apr 08 2015

E.g.f.: cosh(x+y)/cosh(2*(x+y))*exp(x).

T(n,m) = Sum_{k=floor(m/2)..floor(n/2)} Sum_{i=0..k} 4^i*E(2*i)*C(2*k,2*i)*C(n-m,2*k-m), where E(n) are the Euler secant numbers A122045.

A371687 Triangle read by rows: T(n, k) = (-1)^(n-k) * (2n + 1)! [y^(2k)] [x^(2n+1)] arctan(sec(xy)tanh(x)).

Original entry on oeis.org

1, 4, 3, 80, 80, 25, 3904, 5376, 2660, 427, 354560, 626688, 433440, 131712, 12465, 51733504, 111738880, 99242880, 43804992, 9021540, 555731, 11070525440, 28258074624, 30647302400, 17666508288, 5509286640, 816337808, 35135945

Offset: 0

Peter Luschny, Apr 03 2024

Expansion of the exponential generating function arctan(sec(x*y)*tanh(x)), nonzero terms only.

			Triangle starts:
  [0]        1;
  [1]        4,         3;
  [2]       80,        80,       25;
  [3]     3904,      5376,     2660,      427;
  [4]   354560,    626688,   433440,   131712,   12465;
  [5] 51733504, 111738880, 99242880, 43804992, 9021540, 555731;

Cf. A002436 (column 0), A009843 (main diagonal), A012798 (row sums), A012835 (alternating row sums).

Cf. A371688.

egf := arctan(sec(x*y)*tanh(x)):
serx := simplify(series(egf, x, 26)): coeffx := n -> n!*coeff(serx, x, n):
seq(print(seq((-1)^(n-k)*coeff(coeffx(2*n+1), y, 2*k), k = 0..n)), n = 0..6);

cp's OEIS Frontend

A000819 E.g.f.: cos(x)^2 / cos(2x) = Sum_{n >= 0} a(n) * x^(2n) / (2n)!.

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A117437 Expansion of e.g.f.: exp(x)*sec(2*x).

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A291260 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 - 2^k*x/(1 - 4^k*x/(1 - 6^k*x/(1 - 8^k*x/(1 - 10^k*x/(1 - ...)))))).

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A138076 Triangle read by rows: T(n, k) = (-1)^(n+k) * A060187(n+1, k+1).

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Magma

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A335257 Numerators of expansion of arctanh(tan(x)) (odd powers only).

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A371688 Triangle read by rows: T(n, k) = (2*n + 1)! * [y^(2*k)] [x^(2*n+1)] arctan(sec(x*y)*sinh(x)).

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Maple

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A256679 A signed triangle of V. I. Arnold for the Springer numbers (A001586).

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A117437 Expansion of e.g.f.: exp(x)sec(2x).

A291260 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 - 2^kx/(1 - 4^kx/(1 - 6^kx/(1 - 8^kx/(1 - 10^k*x/(1 - ...)))))).

A371688 Triangle read by rows: T(n, k) = (2n + 1)! [y^(2k)] [x^(2n+1)] arctan(sec(xy)sinh(x)).

A371687 Triangle read by rows: T(n, k) = (-1)^(n-k) * (2n + 1)! [y^(2k)] [x^(2n+1)] arctan(sec(xy)tanh(x)).