A186763 -id:A186763 - cp's OEIS frontend

A009628 Expansion of e.g.f.: sinh(x)/(1+x).

0, 1, -2, 7, -28, 141, -846, 5923, -47384, 426457, -4264570, 46910271, -562923252, 7318002277, -102452031878, 1536780478171, -24588487650736, 418004290062513, -7524077221125234, 142957467201379447, -2859149344027588940, 60042136224579367741

Offset: 0

R. H. Hardin

(-1)^n*(A000166 + A000522)/2 = A009179, (-1)^n*(A000166-A000522)/2 = this_sequence.

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A000166, A000522, A009179, A051397, A087208, A186763.

G(x):= sinh(x)/(1+x): f[0]:=G(x): for n from 1 to 21 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..20); # Zerinvary Lajos, Apr 03 2009

a[n_] := (-1)^n (Exp[-1] Gamma[1 + n, -1] - Exp[1] Gamma[1 + n, 1])/2;
Table[a[n], {n, 0, 20}] (* Peter Luschny, Dec 18 2017 *)
With[{nn=30},CoefficientList[Series[Sinh[x]/(1+x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 19 2023 *)

a(n) = n!*polcoeff((sinh(x)/(1+x) + x * O(x^n)), n) \\ Charles R Greathouse IV, Sep 09 2016

x='x+O('x^99); concat([0], Vec(serlaplace(sinh(x)/(1+x)))) \\ Altug Alkan, Dec 18 2017

def A009628(n)
  a = 0
  (0..n).map{|i| a = -i * a + i % 2}
end # Seiichi Manyama, Sep 09 2016

a(n) = (-1)^(n+1)*floor(n!*sinh(1)), n>=1. - Vladeta Jovovic, Aug 10 2002
Let u(1) = 1, u(n) = n*u(n-1) + n (mod 2); then for n>0, a(n) = (-1)^(n+1)*u(n). - Benoit Cloitre, Jan 12 2003
Unsigned sequence satisfies a(n) = n*a(n-1)+a(n-2)-(n-2)*a(n-3), with E.g.f. sinh(z)/(1-z). - Mario Catalani (mario.catalani(AT)unito.it), Feb 08 2003
a(n) = (-1)^(n+1) * n! * Sum_{k=1..floor((n+1)/2)} 1/(2*k-1)!.
a(n) = -n*a(n-1) + n (mod 2). - Seiichi Manyama, Sep 09 2016
a(n) = (-1)^n*(exp(-1)*Gamma(1+n,-1) - exp(1)*Gamma(1+n,1))/2. - Peter Luschny, Dec 18 2017

Extended with signs by Olivier Gérard, Mar 15 1997

Definition clarified by Harvey P. Dale, Mar 19 2023

A051397 a(n) = (2n-2)(2n-1)a(n-1)+1.

Original entry on oeis.org

0, 1, 7, 141, 5923, 426457, 46910271, 7318002277, 1536780478171, 418004290062513, 142957467201379447, 60042136224579367741, 30381320929637160076947, 18228792557782296046168201, 12796612375563171824410077103, 10390849248957295521420982607637

Offset: 0

Aleksandar Petojevic

Harvey P. Dale, Table of n, a(n) for n = 0..225
Romeo Mestrovic, Variations of Kurepa's left factorial hypothesis, arXiv preprint arXiv:1312.7037 [math.NT], 2013.
Romeo Mestrovic, The Kurepa-Vandermonde matrices arising from Kurepa's left factorial hypothesis, Filomat 29:10 (2015), 2207-2215; DOI 10.2298/FIL1510207M.
Aleksandar Petojevic, On Kurepa's Hypothesis for the Left Factorial, FILOMAT (Nis), 12:1 (1998), p. 29-37.

Bisection of abs(A009628). Also bisection of A087208 and of A186763. Cf. A073742, A074790, A275651.

nxt[{n_,a_}]:={n+1,(2(n+1)-2)(2(n+1)-1)a+1}; Transpose[NestList[nxt,{0,0},20]][[2]] (* Harvey P. Dale, Jun 13 2016 *)

a(n) = Sum_{k=0..n-1} (2*n-1)!/(2*k+1)!. a(n) = floor((2*n-1)!*sinh(1)). - Vladeta Jovovic, Aug 10 2002
Conjecture: a(n) +(-4*n^2+6*n-3)*a(n-1) +2*(2*n-3)*(n-2)*a(n-2)=0. - R. J. Mathar, Jan 31 2014
From Peter Bala, Sep 02 2016: (Start)
G.f. sinh(x)/(1 - x^2) = x + 7*x^3/3! + 141*x^5/5! + 5923*x^7/7! + ....
Mathar's conjectured recurrence a(n) = (4*n^2 - 6*n + 3)*a(n-1) - (2*n - 3)*(2*n - 4)*a(n-2) follows easily from the defining recurrence. The sequence b(n) := (2*n - 1)! also satisfies Mathar's recurrence but with b(1) = 1, b(2) = 6. This leads to the continued fraction representation a(n) = (2*n - 1)!*(1 + 1/(6 - 6/(21 - 20/(43 - ... - (2*n - 3)*(2*n - 4)/(4*n^2 - 6*n + 3) )))) for n >= 3. Taking the limit gives the continued fraction representation sinh(1) = A073742 = 1 + 1/(6 - 6/(21 - 20/(43 - ... - (2*n - 3)*(2*n - 4)/((4*n^2 - 6*n + 3) - ... )))). (End)

A186761 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k increasing odd cycles (0<=k<=n). A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 4, 0, 1, 9, 4, 10, 0, 1, 33, 56, 10, 20, 0, 1, 235, 218, 211, 20, 35, 0, 1, 1517, 1982, 833, 616, 35, 56, 0, 1, 12593, 14040, 9612, 2408, 1526, 56, 84, 0, 1, 111465, 134248, 72588, 35176, 5838, 3360, 84, 120, 0, 1, 1122819, 1305126, 797461, 276120, 107710, 12516, 6762, 120, 165, 0, 1

Offset: 0

Emeric Deutsch, Feb 27 2011

Sum of entries in row n is n!.
T(n,0) = A186762(n).
Sum_{k=0..n} k*T(n,k) = A186763(n).

			T(3,1)=4 because we have (1)(23), (12)(3), (13)(2), and (123).
T(4,1)=4 because we have (1)(243), (143)(2), (142)(3), and (132)(4).
Triangle starts:
   1;
   0,  1;
   1,  0,  1;
   1,  4,  0,  1;
   9,  4, 10,  0, 1;
  33, 56, 10, 20, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..170, flattened

Cf. A186762, A186763, A186764, A186766, A186769.

g := exp((t-1)*sinh(z))/(1-z): gser := simplify(series(g, z = 0, 13)): for n from 0 to 10 do P[n] := sort(expand(factorial(n)*coeff(gser, z, n))) end do: for n from 0 to 10 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(b(n-j)*(
      `if`(j::odd, x-1, 0)+(j-1)!)*binomial(n-1, j-1), j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..14);  # Alois P. Heinz, May 12 2017

b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n - j]*(If[OddQ[j], x - 1, 0] + (j - 1)!)*Binomial[n - 1, j - 1], {j, 1, n}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jun 05 2018, after Alois P. Heinz *)

E.g.f.: G(t,z) = exp((t-1)*sinh z)/(1-z).
The 5-variate e.g.f. H(x,y,u,v,z) of permutations with respect to size (marked by z), number of increasing odd cycles (marked by x), number of increasing even cycles (marked by y), number of nonincreasing odd cycles (marked by u), and number of nonincreasing even cycles (marked by v), is given by
H(x,y,u,v,z)=exp(((x-u)sinh z + (y-v)(cosh z - 1))*(1+z)^{(u-v)/2}/(1-z)^{(u+v)/2}.

A186762 Number of permutations of {1,2,...,n} having no increasing odd cycles. A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)

Original entry on oeis.org

1, 0, 1, 1, 9, 33, 235, 1517, 12593, 111465, 1122819, 12313409, 147949593, 1922353925, 26918452691, 403744456541, 6460109224801, 109820584161393, 1976779056442179, 37558742545087481, 751175283283221129, 15774677696321630525, 347042934659313999539, 7981987292809647817237

Offset: 0

Emeric Deutsch, Feb 27 2011

nonn

a(n) = A186761(n,0).

			a(3)=1 because we have (132).
a(4)=9 because we have (12)(34), (13)(24), (14)(23), and the six cyclic permutations of {1,2,3,4}.

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A186761, A186763, A186764, A186765, A186766, A186769.

g := exp(-sinh(z))/(1-z): gser := series(g, z = 0, 27): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 23);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
     ((j-1)!-irem(j, 2))*binomial(n-1, j-1), j=1..n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, May 04 2023

CoefficientList[Series[E^(-Sinh[x])/(1-x), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Mar 16 2014 *)

E.g.f.: g(z) = exp(-sinh z)/(1-z).

a(n) ~ exp(-sinh(1)) * n! = 0.308756853522... * n!. - Vaclav Kotesovec, Mar 16 2014

A184958 Number of nonincreasing even cycles in all permutations of {1,2,...,n}. A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)

Original entry on oeis.org

0, 0, 0, 0, 5, 25, 269, 1883, 20103, 180927, 2172149, 23893639, 326640467, 4246326071, 65675585793, 985133786895, 17069814958319, 290186854291423, 5579050805341613, 106001965301490647, 2241684406438644939, 47075372535211543719

Offset: 0

Emeric Deutsch, Feb 27 2011

nonn

			a(4) = 5 because the only permutations of {1,2,3,4} having nonincreasing even cycles are (1243), (1324), (1342), (1423), and (1432).

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A186761, A186763, A186764, A186766, A186768, A186769.

g := (1/2*(2*(1-cosh(z))-ln(1-z^2)))/(1-z): gser := series(g, z = 0, 27): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 21);

With[{nn=30},CoefficientList[Series[1/2(2(1-Cosh[x])-Log[1-x^2])/(1-x), {x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Oct 22 2011 *)
Table[(n! HarmonicNumber[n] - HypergeometricPFQ[{1, -n}, {}, -1] + (-1)^(n + 1) HypergeometricPFQ[{1, -n}, {}, 1] + n! (2 + (-1)^n LerchPhi[-1, 1, 1 + n] - Log[2]))/2, {n, 0, 20}] (* Benedict W. J. Irwin, May 30 2016 *)

a(n) = Sum_{k>=0} k*A186769(n,k).
E.g.f.: (1/2) * (2*(1-cosh(z)) - log(1-z^2))/(1-z).
a(n) ~ n!/2 * (log(n/2) - 1/exp(1) + 2 - exp(1) + gamma), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 05 2013
a(n) = (n!*H(n)-2F0(1,-n;;-1) + (-1)^(n+1)*2F0(1,-n;;1)+n!*(2+(-1)^n*LerchPhi(-1,1,n+1)-log(2)))/2, where H(n) is the n-th harmonic number. - Benedict W. J. Irwin, May 30 2016

A186767 Number of permutations of {1,2,...,n} having no nonincreasing odd cycles. A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)

Original entry on oeis.org

1, 1, 2, 5, 20, 77, 472, 2585, 21968, 157113, 1724064, 15229645, 204738624, 2151199429, 34194201472, 416221515169, 7631627843840, 105565890206193, 2192501224174080, 33962775502534165, 787900686999286784, 13509825183288167869

Offset: 0

Emeric Deutsch, Feb 27 2011

nonn

a(n) = A186766(n,0).

			a(3)=5 because we have (1)(2)(3), (1)(23), (12)(3), (13)(2), and (123).

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A186761, A186762, A186763, A186764, A186765, A186766, A186769, A186770.

g := exp(sinh(z))/sqrt(1-z^2): gser := series(g, z = 0, 27): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 21);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      binomial(n-1, j-1)*`if`(j::even, (j-1)!, 1), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 13 2017

a[n_] := a[n] = If[n==0, 1, Sum[a[n-j]*Binomial[n-1, j-1]*If[EvenQ[j], (j-1)!, 1], {j, 1, n}]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 18 2017, after Alois P. Heinz *)

E.g.f.: g(z) = exp(sinh z)/sqrt(1-z^2).

A186768 Number of nonincreasing odd cycles in all permutations of {1,2,...,n}. A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)

Original entry on oeis.org

0, 0, 0, 1, 4, 43, 258, 2525, 20200, 222119, 2221190, 28061889, 336742668, 4856656283, 67993187962, 1107076110629, 17713217770064, 322047491979087, 5796854855623566, 116542615962575753, 2330852319251515060, 51380800712458456259

Offset: 0

Emeric Deutsch, Feb 27 2011

nonn

a(n) = Sum(k*A186766(n,k), k>=0).

			a(3)=1 because in (1)(2)(3), (1)(23), (12)(3), (13)(2), (123), and (132) we have a total of 0+0+0+0+0+1 =1 increasing odd cycles.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A186761, A186763, A186764, A186766, A186769, A184958.

g := ((ln((1+z)/(1-z))-2*sinh(z))*1/2)/(1-z): gser := series(g, z = 0, 27): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 21);

CoefficientList[Series[(Log[(1+x)/(1-x)]-2*Sinh[x])/(2*(1-x)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 07 2013 *)

E.g.f.: g(z)=[log((1+z)/(1-z))-2sinh(z)]/(2(1-z)).

a(n) ~ n!/2 * (log(2*n) + gamma - exp(1) + exp(-1)), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 07 2013

Typo in e.g.f. corrected by Vaclav Kotesovec, Oct 07 2013

A296727 Expansion of e.g.f. arcsinh(x)/(1 - x).

Original entry on oeis.org

0, 1, 2, 5, 20, 109, 654, 4353, 34824, 324441, 3244410, 34795485, 417545820, 5536151685, 77506123590, 1144330385625, 18309286170000, 315366695240625, 5676600514331250, 106667957800963125, 2133359156019262500, 45229212438054868125, 995042673637207098750, 22696937952367956440625

Offset: 0

Ilya Gutkovskiy, Dec 19 2017

nonn

			arcsinh(x)/(1 - x) = x/1! + 2*x^2/2! + 5*x^3/3! + 20*x^4/4! + 109*x^5/5! + ...

Cf. A001818, A009628, A081358, A186763, A281964, A296726.

a:=series(arcsinh(x)/(1 - x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019

nmax = 23; CoefficientList[Series[ArcSinh[x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Log[x + Sqrt[1 + x^2]]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

first(n) = x='x+O('x^n); Vec(serlaplace(asinh(x)/(1 - x)), -n) \\ Iain Fox, Dec 19 2017

E.g.f.: log(x + sqrt(1 + x^2))/(1 - x).

a(n) ~ n! * log(1 + sqrt(2)). - Vaclav Kotesovec, Dec 20 2017

A306150 Row sums of A306015.

Original entry on oeis.org

0, 2, 4, 14, 56, 282, 1692, 11846, 94768, 852914, 8529140, 93820542, 1125846504, 14636004554, 204904063756, 3073560956342, 49176975301472, 836008580125026, 15048154442250468, 285914934402758894, 5718298688055177880, 120084272449158735482, 2641853993881492180604

Offset: 0

Peter Luschny, Jun 23 2018

nonn

a(n) is the number of nonderangements of size n in which each fixed point is colored red or blue. For example, with n = 3, the derangements are 231 and 312 and they don't count, the permutations 132, 321, 213 each have 1 fixed point and hence 2 colorings, and the identity 123 with 3 fixed points has 8 colorings, yielding a(3) = 3*2 + 8 = 14 colorings altogether. - David Callan, Dec 19 2021

G. C. Greubel, Table of n, a(n) for n = 0..448

Cf. A000166, A000522, A009628, A186763.

Cf. A306015.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:= [0] cat Coefficients(R!(2*Sinh(x)/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jul 18 2018

egf := 2*sinh(x)/(1-x): ser := series(egf,x,24):
seq(n!*coeff(ser,x,n), n=0..22);

Table[Exp[1] Gamma[n+1, 1] - Subfactorial[n], {n, 0, 22}]
With[{nmax = 50}, CoefficientList[Series[2*Sinh[x]/(1 - x), {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Jul 18 2018 *)

x='x+O('x^30); concat([0], Vec(serlaplace(2*sinh(x)/(1 - x)))) \\ G. C. Greubel, Jul 18 2018

@cached_function
def a(n):
    if n<3: return 2*n
    return n*a(n-1)+a(n-2)-(n-2)*a(n-3)
[a(n) for n in (0..22)]

a(n) = e * Gamma(n + 1, 1) - !(n).
a(n) = Gamma(n + 1, 1) * e - Gamma(n + 1, -1) / e.
a(n) = n*a(n-1) + a(n-2) - (n-2)*a(n-3) for n >= 3.
a(n) = n! [x^n] 2*sinh(x)/(1-x).
a(n) = 2*A186763(n) = (-1)^(n+1)*2*A009628(n) = A000522(n) - A000166(n).

A195326 Numerators of fractions leading to e - 1/e (A174548).

Original entry on oeis.org

0, 2, 2, 7, 7, 47, 47, 5923, 5923, 426457, 426457, 15636757, 15636757, 7318002277, 7318002277, 1536780478171, 1536780478171, 603180793741, 603180793741, 142957467201379447, 142957467201379447

Offset: 0

Paul Curtz, Oct 12 2011

The sequence of approximations of exp(1) obtained by truncating the Taylor series of exp(x) after n terms is A061354(n)/A061355(n) = 1, 2, 5/2, 8/3, 65/24, ...
A Taylor series of exp(-1) is 1, 0, 1/2, 1/3, 3/8, ... and (apart from the first 2 terms) given by A000255(n)/A001048(n). Subtracting both sequences term by term we obtain a series for exp(1) - exp(-1) = 0, 2, 2, 7/3, 7/3, 47/20, 47/20, 5923/2520, 5923/2520, 426457/181440, 426457/181440, ... which defines the numerators here.
Each second of the denominators (that is 3, 2520, 19958400, ...) is found in A085990 (where each third term, that is 60, 19958400, ...) is to be omitted.
This numerator sequence here is basically obtained by doubling entries of A051397, A009628, A087208, or A186763, caused by the standard associations between cosh(x), sinh(x) and exp(x).

			a(0) =  1  -  1;
a(1) =  2  -  0;
a(2) = 5/2 - 1/2.

Cf. A001113, A068985, A197222, A197223.

taylExp1 := proc(n)
        add(1/j!,j=0..n) ;
end proc:
A000255 := proc(n)
        if n <=1 then
                1;
        else
                n*procname(n-1)+(n-1)*procname(n-2) ;
        end if;
end proc:
A001048 := proc(n)
        n!+(n-1)! ;
end proc:
A195326 := proc(n)
        if n = 0 then
                0;
        elif n =1 then
                2;
        else
                taylExp1(n) -A000255(n-2)/A001048(n-1);
        end if;
          numer(%);
end proc:
seq(A195326(n),n=0..20) ; # R. J. Mathar, Oct 14 2011

Material meant to be placed in other sequences removed by R. J. Mathar, Oct 14 2011

cp's OEIS Frontend

A009628 Expansion of e.g.f.: sinh(x)/(1+x).

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A051397 a(n) = (2*n-2)*(2*n-1)*a(n-1)+1.

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A186761 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k increasing odd cycles (0<=k<=n). A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)

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A186762 Number of permutations of {1,2,...,n} having no increasing odd cycles. A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)

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A184958 Number of nonincreasing even cycles in all permutations of {1,2,...,n}. A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)

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A186767 Number of permutations of {1,2,...,n} having no nonincreasing odd cycles. A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)

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A186768 Number of nonincreasing odd cycles in all permutations of {1,2,...,n}. A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)

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A051397 a(n) = (2n-2)(2n-1)a(n-1)+1.