A110501 -id:A110501 - cp's OEIS frontend

A296839 Expansion of e.g.f. tan(x*tan(x/2)) (even powers only).

0, 1, 1, 33, 437, 22205, 978873, 81005113, 7356832669, 949918117653, 142805534055905, 27120922891214801, 6016195462632487941, 1592800634594574194413, 486576430503128985793417, 171866951067212728072402665, 69025662074064538734826793453

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

nonn

			tan(x*tan(x/2)) = x^2/2! + x^4/4! + 33*x^6/6! + 437*x^8/8! + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A000182, A001469, A003718, A009707, A024265, A110501, A296835, A296841, A296842.

nmax = 16; Table[(CoefficientList[Series[Tan[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)] tan(x*tan(x/2)).

a(n) ~ c * d^n * n^(2*n + 1/2) / exp(2*n), where d = 16/Pi^2 = 1.621138938277404343102071411355642222469740394755... is the root of the equation tan(1/sqrt(d)) = Pi*sqrt(d)/4 and c = 1.75568815831... - Vaclav Kotesovec, Dec 21 2017, updated Mar 16 2024

A009752 Expansion of e.g.f. tan(x)*x (even powers only).

Original entry on oeis.org

0, 2, 8, 96, 2176, 79360, 4245504, 313155584, 30460116992, 3777576173568, 581777702256640, 108932957168730112, 24370173276164456448, 6419958484945407574016, 1967044844910430876860416, 693575525634287935244206080, 278846808228005417477465964544, 126799861926498005417315327279104

Offset: 0

R. H. Hardin

nonn

			2*x/(1+e^(2*x)) = 0 + x - 2/2!*x^2 + 8/4!*x^4 - 96/6!*x^6 + 2176/8!*x^8 ...

G. C. Greubel, Table of n, a(n) for n = 0..240
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. 12.

A diagonal of A232933.

Cf. A009725, A024255, A065619, A099028, A110501, A117513.

a := n -> 4^n*n*`if`(n=0,0,abs(euler(2*n-1, 0))): # Peter Luschny, Jun 09 2016

nn = 30; t = Range[0, nn]! CoefficientList[Series[x*Tan[x], {x, 0, nn}], x]; Take[t, {1, nn + 1, 2}] (* T. D. Noe, Sep 20 2012 *)
Table[(-1)^n 4 n PolyLog[1 - 2 n, -I], {n, 0, 19}] (* Peter Luschny, Aug 17 2021 *)

my(x='x+O('x^50)); v=Vec(serlaplace(x*tan(x))); concat([0], vector(#v\2,n,v[2*n-1])) \\ G. C. Greubel, Feb 12 2018

a(n) = n 4^n |E_{2n-1}(1/2)+E_{2n-1}(1)| for n > 0; E_{n}(x) Euler polynomials. - Peter Luschny, Nov 25 2010
a(n) = (2*n)! * [x^(2*n)] tan(x)*x.
a(n) = 2*(2*n)!*Pi^(-2*n)*(4^n-1)*Li{2*n}(1) for n > 0. - Peter Luschny, Jun 29 2012
E.g.f.: sqrt(x)*tan(sqrt(x))= sum(n>=0,  a(n)*x^n/(2*n)! ) = x/T(0) where T(k)= 1 - 4*k^2 + x*(1 - 4*k^2)/T(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Sep 19 2012
E.g.f.: -1 - x^(1/2)- Q(0),where Q(k) = 4*k -1 - x/( 1 - x/ (4*k+1 + x/( 1 + x/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2013
From Peter Luschny, Jun 09 2016: (Start)
a(n) = (4^n-16^n)*Sum_{k=0..2*n} (-1)^(n-k)*Stirling2(2*n, k)*k!/(k+1).
2*a(n)/4^n = A110501(n) for n>=1.
a(n) / 2^n = A117513(n) for n>=1. (End)
a(n) ~ (4*(4^(2*n)-2^(2*n)))*Pi*(n/(Pi*e))^(2*n+1/2)*exp(1/2+1/(24*n)-1/(2880*n^3) +1/(40320*n^5)-...). - Peter Luschny, Jan 16 2017
a(n) = (-1)^n*4*n*PolyLog(1 - 2*n, -i). - Peter Luschny, Aug 17 2021
a(n) = 2*A024255(n). - Alois P. Heinz, Aug 17 2021

Extended and signs tested by Olivier Gérard, Mar 15 1997

A296841 Expansion of e.g.f. sin(x*tan(x/2)) (even powers only).

Original entry on oeis.org

0, 1, 1, -12, -193, -2365, -18552, 500689, 48649969, 2981261772, 169237306055, 9187565146331, 427287357700176, 6011297159973313, -2887128048794477663, -711942625068679870620, -132369975517302093882097, -22968753773651295426439021

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

sign

			sin(x*tan(x/2)) = x^2/2! + x^4/4! - 12*x^6/6! - 193*x^8/8! - 2365*x^10/10! - 18552*x^12/12! + ...

Cf. A001469, A003706, A009515, A110501, A296839, A296842.

nmax = 17; Table[(CoefficientList[Series[Sin[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)] sin(x*tan(x/2)).

A296842 Expansion of e.g.f. cos(x*tan(x/2)) (even powers only).

Original entry on oeis.org

1, 0, -3, -15, -14, 1755, 60357, 1740284, 45816165, 776485557, -37342503290, -7203185712261, -822818831400759, -85463040449605000, -8640073895507612019, -843669753827174738535, -73050419139737972150438, -3478007209663880122501701

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

sign

			cos(x*tan(x/2)) = 1 - 3*x^4/4! - 15*x^6/6! - 14*x^8/8! + 1755*x^10/10! + 60357*x^12/12! + ...

Cf. A001469, A003710, A009080, A110501, A296839, A296841.

nmax = 17; Table[(CoefficientList[Series[Cos[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)] cos(x*tan(x/2)).

A296853 Expansion of e.g.f. tanh(x*tan(x/2)) (even powers only).

Original entry on oeis.org

0, 1, 1, -27, -403, 8345, 688473, -208019, -3189211931, -162605047455, 28806493001105, 5257860587364341, -288068264497990179, -230932276247139756887, -14420179324444754436023, 13944106915630111553887485, 3643613240568912544562868053

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

sign

			tanh(x*tan(x/2)) = x^2/2! + x^4/4! - 27*x^6/6! - 403*x^8/8! + 8345*x^10/10! + ...

Cf. A000182, A001469, A003721, A009817, A110501, A296839, A296841, A296842, A296854, A296856.

nmax = 16; Table[(CoefficientList[Series[Tanh[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)] tanh(x*tan(x/2)).

A296854 Expansion of e.g.f. sinh(x*tan(x/2)) (even powers only).

Original entry on oeis.org

0, 1, 1, 18, 227, 4565, 126648, 4620805, 213569269, 12165013026, 835868220455, 68093897815361, 6483538063860336, 712877916658802713, 89586864207214060057, 12753583150716684461970, 2040805972702652020364603, 364567588100855831300341565

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

nonn

			sinh(x*tan(x/2)) = x^2/2! + x^4/4! + 18*x^6/6! + 227*x^8/8! + 4565*x^10/10! + ...

Cf. A001469, A003717, A009611, A110501, A296839, A296841, A296842, A296853, A296856.

nmax = 17; Table[(CoefficientList[Series[Sinh[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)] sinh(x*tan(x/2)).

A296856 Expansion of e.g.f. cosh(x*tan(x/2)) (even powers only).

Original entry on oeis.org

1, 0, 3, 15, 224, 4545, 126753, 4626076, 213703095, 12167727543, 835893746300, 68091766034061, 6483302813035857, 712860388963255000, 89585739948801890619, 12753524767335858733935, 2040804997678590563632568, 364567987004433619078313961

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

nonn

			cosh(x*tan(x/2)) = 1 + 3*x^4/4! + 15*x^6/6! + 224*x^8/8! + 4545*x^10/10! + 126753*x^12/12! + ...

Cf. A001469, A003711, A009166, A110501, A296839, A296841, A296842, A296853, A296854.

nmax = 17; Table[(CoefficientList[Series[Cosh[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)] cosh(x*tan(x/2)).

A117513 Number of ways of arranging 2*n tokens in a row, with 2 copies of each token from 1 through n, such that between every pair of tokens labeled i (i = 1..n-1) there is exactly one taken labeled i+1.

Original entry on oeis.org

1, 2, 12, 136, 2480, 66336, 2446528, 118984832, 7378078464, 568142287360, 53189920492544, 5949749335001088, 783686338494312448, 120058889459865165824, 21166245289132322242560, 4254864627502524070395904, 967406173145278971994898432, 247007221085479721384365129728

Offset: 1

Nan Zang (nzang(AT)cs.ucsd.edu), Apr 28 2006

nonn

From Paul Barry, Oct 12 2009: (Start)
The aerated sequence is (2^(n/2 - 1) + 0^(n/2)/2)*((1 + (-1)^n)/2)*n!*[x^n](1 + x*tan(x/2)).
Multiples of the unsigned Genocchi numbers A110501: (1, 1, 3, 17, 155,...)*(1, 2, 4, 8, 16,...). (End)

Michael De Vlieger, Table of n, a(n) for n = 1..258
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]

Cf. A002105, A110501, A117514, A117515.

a := n -> (-2)^n*(1 - 2^(2*n))*bernoulli(2*n);
seq(a(n), n = 1..18); # Peter Luschny, Jul 26 2021

Array[(-2)^#*(1 - 2^(2 #))*BernoulliB[2 #] &, 18] (* Michael De Vlieger, Jul 26 2021 *)

# Algorithm of L. Seidel (1877)
# n -> [a(1), ..., a(n)] for n >= 1.
def A117513_list(n) :
    D = [0]*(n+2); D[1] = 1
    R = []; z = 1/2; b = True
    for i in(0..2*n-1) :
        h = i//2 + 1
        if b :
            for k in range(h-1, 0, -1) : D[k] += D[k+1]
            z *= 2
        else :
            for k in range(1, h+1, 1) :  D[k] += D[k-1]
        b = not b
        if b : R.append(D[h]*z)
    return R
A117513_list(15) # Peter Luschny, Jun 29 2012

G.f.: 1/(1-2*x/(1-4*x/(1-8*x/(1-12*x/(1-18*x/(1-24*x/(1-32*x/(1-.../(1-2* floor((n+2)^2/4)*x/(1-... (continued fraction). - Paul Barry, Dec 03 2009
G.f.: T(0), where T(k) = 1 - x*(2*k+2)*(k+1)/( x*(2*k+2)*(k+1) - 1/( 1 - x*(2*k+2)*(k+2)/( x*(2*k+2)*(k+2) - 1/T(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Oct 24 2013
a(n) = (-2)^n*(1 - 2^(2*n))*Bernoulli(2*n). - Peter Luschny, Jul 26 2021

More terms from Paul Barry, Oct 12 2009

A211183 Triangle T(n,k), 0<=k<=n, read by rows, given by (0, 1, 1, 3, 3, 6, 6, 10, 10, 15, ...) DELTA (1, 0, 2, 0, 3, 0, 4, 0, 5, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 4, 1, 0, 7, 19, 11, 1, 0, 38, 123, 107, 26, 1, 0, 295, 1076, 1195, 474, 57, 1, 0, 3098, 12350, 16198, 8668, 1836, 120, 1, 0, 42271, 180729, 268015, 176091, 52831, 6549, 247, 1, 0, 726734, 3290353, 5369639, 4105015, 1564817, 287473, 22145

Offset: 0

Philippe Deléham, Feb 02 2013

			Triangle begins :
1;
0, 1;
0, 1, 1;
0, 2, 4, 1;
0, 7, 19, 11, 1;
0, 38, 123, 107, 26, 1;
0, 295, 1076, 1195, 474, 57, 1;
0, 3098, 12350, 16198, 8668, 1836, 120, 1;
0, 42271, 180729, 268015, 176091, 52831, 6549, 247, 1;
0, 726734, 3290353, 5369639, 4105015, 1564817, 287473, 22145, 502, 1; ...

Paul D. Hanna, Rows n = 0..31, flattened.

Cf. A000366, A110501, A211194, A221972.

T(n,k)=polcoeff(polcoeff(sum(m=0, n, m!*x^m*prod(k=1, m, (y + (k-1)/2)/(1+(k*y+k*(k-1)/2)*x+x*O(x^n)))), n,x),k,y)
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print()) \\ Paul D. Hanna, Feb 03 2013

Sum_{k, 0<=k<=n}T(n,k)*x^(n-k) = A000012(n), A000366(n+1), A110501(n+1), A211194(n), A221972(n) for x = 0, 1, 2, 3, 4 respectively.
T(n,n-1) = A000295(n).
T(n,1) = A000366(n).
G.f.: A(x,y) = Sum_{n>=0} n! * x^n * Product_{k=1..n} (y + (k-1)/2) / (1 + (k*y + k*(k-1)/2)*x). - Paul D. Hanna, Feb 03 2013

A085707 Triangular array A065547 unsigned and transposed.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 3, 0, 1, 6, 17, 17, 0, 1, 10, 55, 155, 155, 0, 1, 15, 135, 736, 2073, 2073, 0, 1, 21, 280, 2492, 13573, 38227, 38227, 0, 1, 28, 518, 6818, 60605, 330058, 929569, 929569, 0, 1, 36, 882, 16086, 211419, 1879038, 10233219, 28820619

Offset: 0

Philippe Deléham, Jul 19 2003

			1;
1,  0;
1,  1,  0;
1,  3,  3,   0;
1,  6, 17,  17,   0;
1, 10, 55, 155, 155, 0;
...

Louis Comtet, Analyse Combinatoire, PUF, 1970, Tome 2, pp. 98-99.

J. M. Hammersley, An undergraduate exercise in manipulation, Math. Scientist, 14 (1989), 1-23. (Annotated scanned copy)
J. M. Hammersley, An undergraduate exercise in manipulation, Math. Scientist, 14 (1989), 1-23.

Row sums Sum_{k>=0} T(n, k) = A006846(n), values of Hammersley's polynomial p_n(1).
Sum_{k>=0} 2^k*T(n, k) = A005647(n), Salie numbers.
Sum_{k>=0} 3^k*T(n, k) = A094408(n).
Sum_{k>=0} 4^k*T(n, k) = A000364(n), Euler numbers.
Cf. A006846, A005647, A000364, A065547, A000795.
Cf. A006846 A005647 A000364 A065547 A000795 A084938.

h[n_, x_] := Sum[c[k]*x^k, {k, 0, n}]; eq[n_] := SolveAlways[h[n, x*(x-1)] == EulerE[2*n, x], x]; row[n_] := Table[c[k], {k, 0, n}] /. eq[n] // First // Abs // Reverse; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Oct 02 2013 *)

Sum_{k >= 0} (-1/2)^k*T(n, k) = (1/2)^n.
Sum_{k >= 0} (-1/6)^k*T(n, k) = (4^(n+1)- 1)/3*(6^n).
Equals A000035 DELTA [0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, ...], where DELTA is Deléham's operator defined in A084938.
T(n,n-1) = A110501(n), Genocchi numbers of first kind of even index. - Philippe Deléham, Feb 16 2007

cp's OEIS Frontend

A296839 Expansion of e.g.f. tan(x*tan(x/2)) (even powers only).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A009752 Expansion of e.g.f. tan(x)*x (even powers only).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A296841 Expansion of e.g.f. sin(x*tan(x/2)) (even powers only).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A296842 Expansion of e.g.f. cos(x*tan(x/2)) (even powers only).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A296853 Expansion of e.g.f. tanh(x*tan(x/2)) (even powers only).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A296854 Expansion of e.g.f. sinh(x*tan(x/2)) (even powers only).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A296856 Expansion of e.g.f. cosh(x*tan(x/2)) (even powers only).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A117513 Number of ways of arranging 2*n tokens in a row, with 2 copies of each token from 1 through n, such that between every pair of tokens labeled i (i = 1..n-1) there is exactly one taken labeled i+1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Sage