A009752 -id:A009752 - cp's OEIS frontend

A024255 a(0)=0, a(n) = n*E(2n-1) for n >= 1, where E(n) = A000111(n) are the Euler (or up-down) numbers.

0, 1, 4, 48, 1088, 39680, 2122752, 156577792, 15230058496, 1888788086784, 290888851128320, 54466478584365056, 12185086638082228224, 3209979242472703787008, 983522422455215438430208, 346787762817143967622103040, 139423404114002708738732982272

Offset: 0

R. H. Hardin

nonn

Number of cyclically alternating permutations of length 2n. Example: a(2)=4 because we have 1324, 1423, 2314, and 2413 (3412 is alternating but not cyclically alternating).

T. D. Noe, Table of n, a(n) for n = 0..100
Noam D. Elkies, On the sums Sum((4k+1)^(-n),k,-inf,+inf), arXiv:math/0101168 [math.CA], 2001-2003.
Noam D. Elkies, On the sums Sum_{k = -infinity .. infinity} (4k+1)^(-n), Amer. Math. Monthly, 110 (No. 7, 2003), 561-573.
Luis Ferroni, Alejandro H. Morales, and Greta Panova, Skew shapes, Ehrhart positivity and beyond, arXiv:2503.16403 [math.CO], 2025. See p. 16.
G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.

Cf. A000111, A009752.

a := n -> (-1)^n*2^(2*n-1)*(1-2^(2*n))*bernoulli(2*n); # Peter Luschny, Jun 08 2009

nn = 30; t = Range[0, nn]! CoefficientList[Series[Tan[x]*x/2, {x, 0, nn}], x]; Take[t, {1, nn, 2}]
Table[(-1)^n 2 n PolyLog[1 - 2 n, -I], {n, 0, 19}] (* Peter Luschny, Aug 17 2021 *)

from itertools import accumulate, islice, count
def A024255_gen(): # generator of terms
    yield from (0,1)
    blist = (0,1)
    for n in count(2):
        yield n*(blist := tuple(accumulate(reversed(tuple(accumulate(reversed(blist),initial=0))),initial=0)))[-1]
A024255_list = list(islice(A024255_gen(),40)) # Chai Wah Wu, Jun 09-11 2022

a(n) = 2^(n-1)*(2^n-1)*|B_n|.
E.g.f.: tan(x)*x/2 (even part).
a(n) = (2*n)!*Pi^(-2*n)*(4^n-1)*Li{2*n}(1) for n > 0. - Peter Luschny, Jun 29 2012
G.f.: Q(0)*x/(1-4*x), where Q(k) = 1 - 16*x^2*(k+2)*(k+1)^3/( 16*x^2*(k+2)*(k+1)^3 - (1 - 8*x*k^2 - 12*x*k -4*x)*(1 - 8*x*k^2 - 28*x*k -24*x)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 23 2013
a(n) = A009752(n)/2. - Alois P. Heinz, Aug 17 2021
a(n) = (-1)^n*2*n*PolyLog(1 - 2*n, -i). - Peter Luschny, Aug 17 2021

Edited by Emeric Deutsch, Jul 01 2009

A009707 Expansion of e.g.f. tan(tan(x)*x) (even powers only).

Original entry on oeis.org

0, 2, 8, 336, 15616, 1450240, 185032704, 33566984192, 7971973332992, 2424984197529600, 915532582868746240, 420569934453637906432, 230845747512083447021568, 149228982402223336708898816

Offset: 0

R. H. Hardin

nonn

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

Cf. A000182, A003718, A009752, A024265.

nmax = 20; Table[(CoefficientList[Series[Tan[x*Tan[x]], {x, 0, 2*nmax}], x] * Range[0, 2 nmax]!)[[n]], {n, 1, 2*nmax + 1, 2}] (* Vaclav Kotesovec, Dec 21 2017 *)

a(n) ~ c * d^n * n^(2*n + 1/2) / exp(2*n), where d = 3.9786913954409425781217887822690623430980810... is the root of the equation tan(2/sqrt(d)) = Pi*sqrt(d)/4 and c = 1.4057183994645... - Vaclav Kotesovec, Dec 21 2017

Extended and signs tested Mar 15 1997 by Olivier Gérard.

A099028 Euler-Seidel matrix T(k,n) with start sequence e.g.f. 2x/(1+e^(2x)), read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 0, -1, -2, -3, -3, -2, 0, 0, 3, 6, 8, 8, 25, 25, 22, 16, 8, 0, 0, -25, -50, -72, -88, -96, -96, -427, -427, -402, -352, -280, -192, -96, 0, 0, 427, 854, 1256, 1608, 1888, 2080, 2176, 2176, 12465, 12465, 12038, 11184, 9928, 8320, 6432, 4352, 2176, 0

Offset: 0

Ralf Stephan, Sep 27 2004

In an Euler-Seidel matrix, the rows are consecutive pairwise sums and the columns consecutive differences.

			Seidel matrix:
[    0     1    -2     0     8     0   -96     0  2176     0]
[    1    -1    -2     8     8   -96   -96  2176  2176     .]
[    0    -3     6    16   -88  -192  2080  4352     .     .]
[   -3     3    22   -72  -280  1888  6432     .     .     .]
[    0    25   -50  -352  1608  8320     .     .     .     .]
[   25   -25  -402  1256  9928     .     .     .     .     .]
[    0  -427   854 11184     .     .     .     .     .     .]
[ -427   427 12038     .     .     .     .     .     .     .]
[    0 12465     .     .     .     .     .     .     .     .]
[12465     .     .     .     .     .     .     .     .     .]

D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.

First column (odd part) is A009843, main diagonal is in A099029. Antidiagonal sums are in A065619. Cf. A009752.

T[k_, n_] := T[k, n] = If[k == 0, SeriesCoefficient[2x/(1 + E^(2x)), {x, 0, n}] n!, T[k-1, n] + T[k-1, n+1]];
Table[T[k-n, n], {k, 0, 9}, {n, 0, k}] (* Jean-François Alcover, Jun 11 2019 *)

def SeidelMatrixA099028(dim):
    E = matrix(ZZ, dim)
    t = taylor(2*x/(1+exp(2*x)), x, 0, dim + 1)
    for k in (0..dim-1):
        E[0, k] = factorial(k) * t.coefficient(x, k)
    R = [0]
    for n in (1..dim-1):
        for k in (0..dim-n-1):
            E[n, k] = E[n-1, k] + E[n-1, k+1]
        R.extend([E[n-k,k] for k in (0..n)])
    return R
print(SeidelMatrixA099028(10)) # Peter Luschny, Jul 02 2016

Recurrence: T(k, n) = T(k-1, n) + T(k-1, n+1).

A232933 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping, cyclic wrap-around) occurrences of the consecutive step pattern UDU (U=up, D=down); triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

Original entry on oeis.org

1, 1, 0, 2, 3, 3, 12, 4, 8, 35, 45, 40, 144, 348, 132, 96, 910, 1862, 1316, 952, 5976, 11600, 14808, 5760, 2176, 39942, 100260, 123606, 63360, 35712, 306570, 919270, 1069910, 910650, 343040, 79360, 2698223, 8427243, 11694397, 10673641, 4477440, 1945856

Offset: 0

Alois P. Heinz, Dec 02 2013

			T(2,1) = 2: 12, 21 (the two U's of UDU overlap).
T(3,0) = 3: 132, 213, 321.
T(3,1) = 3: 123, 231, 312.
T(4,0) = 12: 1243, 1342, 1432, 2134, 2143, 2431, 3124, 3214, 3421, 4213, 4312, 4321.
T(4,1) = 4: 1234, 2341, 3412, 4123.
T(4,2) = 8: 1324, 1423, 2314, 2413, 3142, 3241, 4132, 4231.
Triangle T(n,k) begins:
:  0 :      1;
:  1 :      1;
:  2 :      0,      2;
:  3 :      3,      3;
:  4 :     12,      4,       8;
:  5 :     35,     45,      40;
:  6 :    144,    348,     132,     96;
:  7 :    910,   1862,    1316,    952;
:  8 :   5976,  11600,   14808,   5760,   2176;
:  9 :  39942, 100260,  123606,  63360,  35712;
: 10 : 306570, 919270, 1069910, 910650, 343040, 79360;

Alois P. Heinz, Rows n = 0..200, flattened

Column k=0 gives A232899.
Row sums give A000142.
T(2n,n) gives A009752(n) = 2n * A000182(n) for n>0.
T(2n+1,n) gives (2n+1) * A024283(n) for n>0.
Cf. A295987.

b:= proc(u, o, t) option remember; `if`(u+o=0,
     `if`(t=2, x, 1), expand(
      add(b(u+j-1, o-j, 2)*`if`(t=3, x, 1), j=1..o)+
      add(b(u-j, o+j-1, `if`(t=2, 3, 1)), j=1..u)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))
        (`if`(n<2, 1, n* b(0, n-1, 1))):
seq(T(n), n=0..12);

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, If[t == 2, x, 1], Expand[Sum[ b[u + j - 1, o - j, 2]*If[t == 3, x, 1], {j, 1, o}] + Sum[b[u - j, o + j - 1, If[t == 2, 3, 1]], {j, 1, u}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]]   [If[n < 2, 1, n*b[0, n - 1, 1]]];
T /@ Range[0, 12] // Flatten (* Jean-François Alcover, Dec 19 2020, after Alois P. Heinz *)

A009725 Expansion of e.g.f.: tan(x)*(1+x).

Original entry on oeis.org

0, 1, 2, 2, 8, 16, 96, 272, 2176, 7936, 79360, 353792, 4245504, 22368256, 313155584, 1903757312, 30460116992, 209865342976, 3777576173568, 29088885112832, 581777702256640, 4951498053124096, 108932957168730112, 1015423886506852352, 24370173276164456448

Offset: 0

R. H. Hardin

a(2n) = A009752(n), a(2n+1) = A000182(n+1).

With[{nn=30},CoefficientList[Series[Tan[x]*(1+x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 29 2020 *)

E.g.f.: x*(1+x)*Q(0), where Q(k) = 1 - x^2/(x^2 - (2*k+1)*(2*k+3)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 16 2013

Definition clarified and prior Mathematica program replaced by Harvey P. Dale, Oct 29 2020

A011248 Twice A000364.

Original entry on oeis.org

2, 2, 10, 122, 2770, 101042, 5405530, 398721962, 38783024290, 4809759350882, 740742376475050, 138697748786275802, 31029068327114173810, 8174145018586247784722, 2504519282807259730936570, 883087786498046209107365642, 355038783159078578873329579330, 161446598471775796124336494906562

Offset: 0

N. J. A. Sloane

nonn

G. Almkvist, Many correct digits of Pi, revisited, Amer. Math. Monthly, 104 (1997), 351-353.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]

Cf. A000364, A009752.

a[n_] := (-1)^n 4 Im[PolyLog[-2 n, I]];
Table[a[n], {n, 0, 17}] (* Peter Luschny, Aug 18 2021 *)

E.g.f.: 2 - 2/Q(0), where Q(k)= 1 - (2*k+1)*(2*k+2)/x + 1/x*(2*k+1)*(2*k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 01 2013
From Peter Luschny, Aug 18 2021: (Start)
a(n) = (-1)^n*4^(2*n+1)*(Bernoulli(2*n+1, 3/4) - Bernoulli(2*n+1, 1/4))/(2*n+1).
a(n) = (-1)^n*4*Im(PolyLog(-2*n, i)). (End)

A243963 a(n) = n4^n(-Z(1-n, 1/4)/2 + Z(1-n, 3/4)/2 - Z(1-n, 1)*(1 - 2^(-n))) for n > 0 and a(0) = 0, where Z(n, c) is the Hurwitz zeta function.

Original entry on oeis.org

0, 0, 2, 3, -8, -25, 96, 427, -2176, -12465, 79360, 555731, -4245504, -35135945, 313155584, 2990414715, -30460116992, -329655706465, 3777576173568, 45692713833379, -581777702256640, -7777794952988025, 108932957168730112, 1595024111042171723, -24370173276164456448

Offset: 0

Paul Curtz, Jun 16 2014

sign

Previous name was: 0 followed by -(n+1)*A163747(n).
Difference table of a(n):
0,    0,    2,     3,    -8,   -25,...
0,    2,    1,   -11,   -17,   121,...
2,   -1,  -12,    -6,   138,   210,...
-3, -11,    6,   144,    72, -3144,...
-8,  17,  138,   -72, -3216, -1608,...
25, 121, -210, -3144,  1608,...
a(n) is an autosequence of second kind. Its inverse binomial transform is the signed sequence. Its main diagonal is the first upper diagonal multiplied by 2.

Cf. A132049, A065619.

a := n -> `if`(n=0, 0, n*4^n*(-Zeta(0, 1-n, 1/4)/2 + Zeta(0, 1-n, 3/4)/2 + Zeta(1-n)*(2^(-n)-1))): seq(a(n), n=0..24); # Peter Luschny, Jul 21 2020

a[0] = 0; a[n_] := -n*SeriesCoefficient[(2*E^x*(1 - E^x))/(1 + E^(2*x)), {x, 0, n-1}]*(n-1)!; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jun 17 2014 *)

a(n) = 0, 0, followed by (period 4: repeat 1, 1, -1, -1)*A065619(n+2).

a(2n) = (-1)^(n+1)A009752(n). a(2n+1) = (-1)^n*A009843(n+1).

New name by Peter Luschny, Jul 21 2020

A296543 Expansion of e.g.f. tanh(exp(x)-1).

Original entry on oeis.org

0, 1, 1, -1, -11, -33, 61, 1367, 7253, -12561, -580499, -4701497, 4669765, 580325215, 6636339165, 1365901495, -1122870368715, -17289945450289, -31110588453299, 3713822629274023, 74717183313957413, 280555705771423039, -19253195126787261507, -496715617694137066089, -3008746115751273626347

Offset: 0

Ilya Gutkovskiy, Dec 15 2017

sign

			tanh(exp(x)-1) = x/1! + x^2/2! - x^3/3! - 11*x^4/4! - 33*x^5/5! + 61*x^6/6! + 1367*x^7/7! + ...

Cf. A009752, A024429, A024430, A080832, A296544.

a:=series(tanh(exp(x)-1),x=0,25): seq(n!*coeff(a,x,n),n=0..24); # Paolo P. Lava, Mar 27 2019

nmax = 24; CoefficientList[Series[Tanh[Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 24; CoefficientList[Series[Sinh[Exp[x] - 1]/Cosh[Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 24; CoefficientList[Series[(Exp[x] - 1)/(1 + ContinuedFractionK[(Exp[x] - 1)^2, 2 k - 1, {k, 2, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: sinh(exp(x)-1)/cosh(exp(x)-1).

E.g.f.: (exp(x)-1)/(1 + (exp(x)-1)^2/(3 + (exp(x)-1)^2/(5 + (exp(x)-1)^2/(7 + (exp(x)-1)^2/(9 + ...))))), a continued fraction.

cp's OEIS Frontend

A024255 a(0)=0, a(n) = n*E(2n-1) for n >= 1, where E(n) = A000111(n) are the Euler (or up-down) numbers.

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A009707 Expansion of e.g.f. tan(tan(x)*x) (even powers only).

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A099028 Euler-Seidel matrix T(k,n) with start sequence e.g.f. 2x/(1+e^(2x)), read by antidiagonals.

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A232933 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping, cyclic wrap-around) occurrences of the consecutive step pattern UDU (U=up, D=down); triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

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A009725 Expansion of e.g.f.: tan(x)*(1+x).

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Extensions

A011248 Twice A000364.

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A243963 a(n) = n*4^n*(-Z(1-n, 1/4)/2 + Z(1-n, 3/4)/2 - Z(1-n, 1)*(1 - 2^(-n))) for n > 0 and a(0) = 0, where Z(n, c) is the Hurwitz zeta function.

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Maple

Mathematica

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A296543 Expansion of e.g.f. tanh(exp(x)-1).

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A243963 a(n) = n4^n(-Z(1-n, 1/4)/2 + Z(1-n, 3/4)/2 - Z(1-n, 1)*(1 - 2^(-n))) for n > 0 and a(0) = 0, where Z(n, c) is the Hurwitz zeta function.