A006229 -id:A006229 - cp's OEIS frontend

A059419 Triangle T(n,k) (1 <= k <= n) of tangent numbers, read by rows: T(n,k) = coefficient of x^n/n! in expansion of (tan x)^k/k!.

1, 0, 1, 2, 0, 1, 0, 8, 0, 1, 16, 0, 20, 0, 1, 0, 136, 0, 40, 0, 1, 272, 0, 616, 0, 70, 0, 1, 0, 3968, 0, 2016, 0, 112, 0, 1, 7936, 0, 28160, 0, 5376, 0, 168, 0, 1, 0, 176896, 0, 135680, 0, 12432, 0, 240, 0, 1, 353792, 0, 1805056, 0, 508640, 0, 25872, 0, 330, 0, 1, 0

Offset: 1

N. J. A. Sloane, Jan 30 2001

(tan(x))^k = sum{n>0, If n+k is odd, T(n,k) = 0 = n!/k!*(-1)^((n+k)/2)*sum{j=k..n} (j!/n!) * Stirling2(n,j) * 2^(n-j) * (-1)^(n+j-k) * binomial(j-1,k-1)*x^n}. - Vladimir Kruchinin, Aug 13 2012

Also the Bell transform of A009006(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

			     1;
     0,     1;
     2,     0,     1;
     0,     8,     0,    1;
    16,     0,    20,    0,    1;
     0,   136,     0,   40,    0,   1;
   272,     0,   616,    0,   70,   0,   1;
     0,  3968,     0, 2016,    0, 112,   0,  1;
  7936,     0, 28160,    0, 5376,   0, 168,  0,  1;

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 259.

Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
Toufik Mansour, Mark Shattuck, Combinatorial parameters on bargraphs of permutations, Transactions on Combinatorics, Article 1, Vol. 7, Issue 2, June 2018, Page 1-16.

Diagonals give A000182, A024283, A059420 (interspersed with 0's), also A007290, A059421. Row sums give A006229. Essentially the same triangle as A008308.

A111593 (signed triangle with extra column k=0 and row n=0).

A059419 := proc(n,k) option remember; if n = k then 1; elif k <0 or k > n then 0; else  procname(n-1,k-1)+k*(k+1)*procname(n-1,k+1) ; end if; end proc: # R. J. Mathar, Feb 11 2011
# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> 2^(n+1)*abs(euler(n+1, 1)), 10); # Peter Luschny, Jan 26 2016

d[f_ ] := (1+x^2)*D[f, x]; d[ f_, n_] := Nest[d, f, n]; row[n_] := Rest[ CoefficientList[ d[Exp[x*t], n] /. x -> 0, t]]; Flatten[ Table[ row[n], {n, 1, 12}]] (* Jean-François Alcover, Dec 21 2011, after Peter Bala *)
rows = 12;
t = Table[2^(n+1)*Abs[EulerE[n+1, 1]], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

T(n,k)=if(k<1 || k>n,0,n!*polcoeff(tan(x+x*O(x^n))^k/k!,n))

def A059419_triangle(dim):
    M = matrix(ZZ, dim, dim)
    for n in (0..dim-1): M[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n,k] = M[n-1,k-1]+(k+1)*(k+2)*M[n-1,k+1]
    return M
A059419_triangle(9) # Peter Luschny, Sep 19 2012

T(n+1, k) = T(n, k-1) + k*(k+1)*T(n, k+1), T(n, n) = 1.
If n+k is odd, T(n,k) = 0 = 1/k!*(-1)^((n+k)/2)*Sum_{j=k..n} j!* Stirling2(n,j)*2^(n-j)*(-1)^(n+j-k)*binomial(j-1,k-1). - Vladimir Kruchinin, Feb 10 2011
E.g.f.: exp(t*tan(x))-1 = t*x + t^2*x^2/2! + (2*t + t^3)*x^3/3! + ....
The row polynomials are given by D^n(exp(x*t)) evaluated at x = 0, where D is the operator (1+x^2)*d/dx. - Peter Bala, Nov 25 2011
The o.g.f.s of the diagonals of this triangle are rational functions obtained from the series reversion (x-t*tan(x))^(-1) = x/(1-t) + 2*t/(1-t)^4*x^3/3! + 8*t*(2+3*t)/(1-t)^7*x^5/5! + 16*t*(17+78*t+45*t^2)/(1-t)^10*x^7/7! + .... For example, the fourth subdiagonal has o.g.f. 8*t*(2+3*t)/(1-t)^7 = 16*t + 136*t^2 + 616*t^3 + .... - Peter Bala, Apr 23 2012
With offset 0 and initial column of zeros, except for T(0,0) = 1, e.g.f.(t,x) = e^(x*tan(t)) = e^(P(.,x)t) ; the lowering operator, L = atan(d/dx) ; and the raising operator, 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). The sequence is a binomial Sheffer sequence. - Tom Copeland, Oct 01 2015

More terms from Larry Reeves (larryr(AT)acm.org), Feb 01 2001

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

A047691 Numerators of coefficients in Taylor series for exp(tan(x)).

Original entry on oeis.org

1, 1, 1, 1, 3, 37, 59, 137, 871, 41641, 325249, 3887, 35797, 241586893, 24362249, 5721418891, 342232522657, 4315903789009, 8224154352439, 2832484672207, 23157229065769, 183184249105857781, 9926476934520521, 2154299222076719401

Offset: 0

N. J. A. Sloane

			1 + 1*x + (1/2)*x^2 + (1/2)*x^3 + (3/8)*x^4 + (37/120)*x^5 + (59/240)*x^6 + (137/720)*x^7 + (871/5760)*x^8 + ...

CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.

Cf. A047692, A006229.

Numerator[CoefficientList[Series[Exp[Tan[x]],{x,0,30}],x]] (* Harvey P. Dale, May 19 2015 *)

A331610 Expansion of e.g.f.: exp(1 / (1 - tan(x)) - 1).

Original entry on oeis.org

1, 1, 3, 15, 97, 777, 7379, 80983, 1007137, 13986289, 214383171, 3593224767, 65347120705, 1281151315641, 26928292883795, 603928982033863, 14392387319349697, 363135896514611041, 9669298448057196291, 270932711729869233903, 7967970654277850949025

Offset: 0

Ilya Gutkovskiy, Jan 22 2020

nonn

Robert Israel, Table of n, a(n) for n = 0..427

Cf. A000111, A000828, A003707, A004211, A006229, A331607.

S:= series(exp(1/(1-tan(x))-1), x, 31):
seq(coeff(S,x,i)*i!, i=0..30); # Robert Israel, Dec 10 2024

nmax = 20; CoefficientList[Series[Exp[1/(1 - Tan[x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
A000111[n_] := If[EvenQ[n], Abs[EulerE[n]], Abs[(2^(n + 1) (2^(n + 1) - 1) BernoulliB[n + 1])/(n + 1)]]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] 2^(k - 1) A000111[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

E.g.f.: exp(sin(x) / (cos(x) - sin(x))).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * 2^(k-1) * A000111(k) * a(n-k).
a(n) ~ 2^(2*n - 1/4) * exp(1/Pi - 1/2 + 2^(3/2)*sqrt(n/Pi) - n) * n^(n - 1/4) / Pi^(n + 1/4). - Vaclav Kotesovec, Jan 27 2020

A047692 Denominators of coefficients in Taylor series for exp(tan(x)).

Original entry on oeis.org

1, 1, 2, 2, 8, 120, 240, 720, 5760, 362880, 3628800, 57600, 691200, 6227020800, 830269440, 261534873600, 20922789888000, 355687428096000, 914624815104000, 426824913715200, 4742499041280000, 51090942171709440000, 3784514234941440000

Offset: 0

N. J. A. Sloane

			1 + 1*x + (1/2)*x^2 + (1/2)*x^3 + (3/8)*x^4 + (37/120)*x^5 + (59/240)*x^6 + (137/720)*x^7 + (871/5760)*x^8 + ...

CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.

Cf. A047691, A006229.

A296835 Expansion of e.g.f. exp(x*tan(x/2)) (even powers only).

Original entry on oeis.org

1, 1, 4, 33, 451, 9110, 253401, 9246881, 427272364, 24332740569, 1671761966755, 136185663849422, 12966840876896193, 1425738305622057713, 179172604156015950676, 25507107918052543195905, 4081610970381242583997171, 729135575105289450378655526

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

nonn

			exp(x*tan(x/2)) = 1 + x^2/2! + 4*x^4/4! + 33*x^6/6! + 451*x^8/8! + ...

Cf. A001469, A003723, A006229, A009252, A009273, A110501, A296836.

nmax = 17; Table[(CoefficientList[Series[Exp[x Tan[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

a(n) = (2*n)! * [x^(2*n)] exp(x*tan(x/2)).

A168404 E.g.f.: Sum_{n>=0} tan(2^n*x)^n/n!.

Original entry on oeis.org

1, 2, 16, 528, 67584, 34210304, 69391122432, 565356426987520, 18478277930015260672, 2419401354886413876592640, 1267940756758206239694099841024, 2658665157828553829995392867121496064

Offset: 0

Paul D. Hanna, Nov 25 2009

nonn

			E.g.f.: A(x) = 1 + 2*x + 16*x^2/2! + 528*x^3/3! + 67584*x^4/4! +...
A(x) = 1 + tan(2*x) + tan(4*x)^2/2! + tan(8*x)^3/3! + tan(16*x)^4/4! +...+ tan(2^n*x)^n/n! +...
a(n) = coefficient of x^n/n! in G(x)^(2^n) where G(x) = exp(tan(x)):
G(x) = 1 + x + x^2/2! + 3*x^3/3! + 9*x^4/4! + 37*x^5/5! + 177*x^6/6! +...+ A006229(n)*x^n/n! +...

Cf. A006229 (exp(tan x)), variants: A136632, A168402, A168403, A168405, A168406, A168407, A168408.

{a(n)=n!*polcoeff(sum(k=0,n,tan(2^k*x +x*O(x^n))^k/k!),n)}

{a(n)=n!*polcoeff(exp(2^n*tan(x +x*O(x^n))),n)}

a(n) = [x^n/n! ] exp(2^n*tan(x)) for n>=0.

A296836 Expansion of e.g.f. exp(x*tanh(x/2)) (even powers only).

Original entry on oeis.org

1, 1, 2, 3, -3, 20, 105, -5271, 133826, -2714517, 25525845, 2131781300, -235250824479, 17527695547713, -1124258412169438, 58383380825728035, -975024061456732035, -398903577787777972396, 97649546210035758250281, -17069419358223320552890167

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

sign

			exp(x*tanh(x/2)) = 1 + x^2/2! + 2*x^4/4! + 3*x^6/6! - 3*x^8/8! + ...

Cf. A001469, A003723, A006229, A009252, A009273, A110501, A296835.

nmax = 19; Table[(CoefficientList[Series[Exp[x Tanh[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

a(n) = (2*n)! * [x^(2*n)] exp(x*tanh(x/2)).

A008308 Triangle of tangent numbers.

Original entry on oeis.org

1, 1, 2, 1, 8, 1, 16, 20, 1, 136, 40, 1, 272, 616, 70, 1, 3968, 2016, 112, 1, 7936, 28160, 5376, 168, 1, 176896, 135680, 12432, 240, 1, 353792, 1805056, 508640, 25872, 330, 1, 11184128, 11977856, 1595264, 49632, 440, 1, 22368256, 154918400, 59835776

Offset: 1

N. J. A. Sloane

			Triangle begins:
    1;
    1;
    2,  1;
    8,  1;
   16, 20, 1;
  136, 40, 1;
  ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 259.

Essentially the same triangle as A059419, which is the main entry for this triangle.

Row sums give A006229.

T[n_, n_] = 1; T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n - 1, k - 1] + k*(k + 1)*T[n - 1, k + 1]; T[, ] = 0;
row[n_] := DeleteCases[Table[T[n, k], {k, 1, n}] , 0];
Array[row, 13] // Flatten (* Jean-François Alcover, Nov 09 2017 *)

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

A013516 Denominators in the Taylor expansion exp(cosec(x)-cot(x))=1 + x/2 + x^2/8 + x^3/16 + 3x^4/128 + 37x^5/3840 + 59*x^6/15360 + ...

Original entry on oeis.org

1, 2, 8, 16, 128, 3840, 15360, 92160, 1474560, 185794560, 3715891200, 117964800, 2831155200, 51011754393600, 13603134504960, 8569974738124800, 1371195958099968000, 46620662575398912000, 239763407530622976000

Offset: 0

Patrick Demichel (patrick.demichel(AT)hp.com)

The numerators are apparently the same as A047691.

			exp(cosec(x)-cot(x)) = 1 +1*x/(2^1*1!) + 1*x^2/(2^2*2!) + 3*x^3/(2^3*3!) + 9*x^4/(2^4*4!) + 37*x^5/(2^5*5!) +  177*x^6/(2^6*6!) +959*x^7/(2^7*7!)+ ...

Cf. A006229, A002425 (expansion of cosec(x)-cot(x)).

A013516 := proc(n)
        exp(csc(x)-cot(x)) ;
        coeftayl( %,x=0,n) ;
        denom(%) ;
end proc:  # R. J. Mathar, Dec 18 2011

a(n) = A047692(n) * 2^n. - Sean A. Irvine, Aug 07 2018

Corrected by R. J. Mathar, Dec 18 2011

Showing 1-10 of 12 results. Next

cp's OEIS Frontend

A059419 Triangle T(n,k) (1 <= k <= n) of tangent numbers, read by rows: T(n,k) = coefficient of x^n/n! in expansion of (tan x)^k/k!.

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A111593 Triangle of tanh numbers.

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A047691 Numerators of coefficients in Taylor series for exp(tan(x)).

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Mathematica

A331610 Expansion of e.g.f.: exp(1 / (1 - tan(x)) - 1).

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Maple

Mathematica

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A047692 Denominators of coefficients in Taylor series for exp(tan(x)).

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A296835 Expansion of e.g.f. exp(x*tan(x/2)) (even powers only).

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Mathematica

Formula

A168404 E.g.f.: Sum_{n>=0} tan(2^n*x)^n/n!.

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PARI

PARI

Formula

A296836 Expansion of e.g.f. exp(x*tanh(x/2)) (even powers only).

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Keywords

A013516 Denominators in the Taylor expansion exp(cosec(x)-cot(x))=1 + x/2 + x^2/8 + x^3/16 + 3x^4/128 + 37x^5/3840 + 59*x^6/15360 + ...