A087208 -id:A087208 - 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)

A330044 Expansion of e.g.f. exp(x) / (1 - x^3).

Original entry on oeis.org

1, 1, 1, 7, 25, 61, 841, 5251, 20497, 423865, 3780721, 20292031, 559501801, 6487717237, 44317795705, 1527439916731, 21798729916321, 180816606476401, 7478345832314977, 126737815733490295, 1236785588298582841, 59677199741873516461, 1171057417377450325801

Offset: 0

Ilya Gutkovskiy, Nov 28 2019

nonn

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A000522, A087208, A100732, A158757, A330045.

[n le 3 select 1 else 1 + 6*Binomial(n-1,3)*Self(n-3): n in [1..41]]; // G. C. Greubel, Dec 05 2021

nmax = 22; CoefficientList[Series[Exp[x]/(1 - x^3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[n!/(n - 3 k)!, {k, 0, Floor[n/3]}], {n, 0, 22}]

[sum(factorial(3*k)*binomial(n, 3*k) for k in (0..n//3)) for n in (0..40)] # G. C. Greubel, Dec 05 2021

G.f.: Sum_{k>=0} (3*k)! * x^(3*k) / (1 - x)^(3*k + 1).
a(0) = a(1) = a(2) = 1; a(n) = n * (n - 1) * (n - 2) * a(n - 3) + 1.
a(n) = Sum_{k=0..floor(n/3)} n! / (n - 3*k)!.
a(n) ~ n! * (exp(1)/3 + 2*cos(sqrt(3)/2 - 2*Pi*n/3) / (3*exp(1/2))). - Vaclav Kotesovec, Apr 18 2020
a(n) = A158757(n, 2*n). - G. C. Greubel, Dec 05 2021

A330045 Expansion of e.g.f. exp(x) / (1 - x^4).

Original entry on oeis.org

1, 1, 1, 1, 25, 121, 361, 841, 42001, 365905, 1819441, 6660721, 498971881, 6278929801, 43710250585, 218205219961, 21795091762081, 358652470233121, 3210080802962401, 20298322381652065, 2534333270094778681, 51516840824285500441, 563561785768079119561

Offset: 0

Ilya Gutkovskiy, Nov 28 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..450

Cf. A000522, A087208, A100733, A330044, A334157.

Outer diagonal of A158777.

nmax = 22; CoefficientList[Series[Exp[x]/(1 - x^4), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[n!/(n - 4 k)!, {k, 0, Floor[n/4]}], {n, 0, 22}]

G.f.: Sum_{k>=0} (4*k)! * x^(4*k) / (1 - x)^(4*k + 1).
a(0) = a(1) = a(2) = a(3) = 1; a(n) = n*(n - 1)*(n - 2)*(n - 3)*a(n - 4) + 1.
a(n) = Sum_{k=0..floor(n/4)} n! / (n - 4*k)!.
a(n) ~ n! * (2*cos(Pi*n/2 - 1) + exp(1) + (-1)^n*exp(-1))/4. - Vaclav Kotesovec, Apr 18 2020

A002747 Logarithmic numbers.

Original entry on oeis.org

1, -2, 9, -28, 185, -846, 7777, -47384, 559953, -4264570, 61594841, -562923252, 9608795209, -102452031878, 2017846993905, -24588487650736, 548854382342177, -7524077221125234, 187708198761024553, -2859149344027588940, 78837443479630312281, -1320926996940746090302

Offset: 1

N. J. A. Sloane

sign

abs(a(n)) is also the number of distinct routes starting from a point A and ending at a point B, without traversing any edge more than once, when there are n bi-directional edges connecting A and B. E.g., if there are 3 edges p, q and r from A to B, then the 9 routes starting from A and ending at B are p, q, r, pqr, prq, rpq, rqp, qpr and qrp. - Nikita Kiran, Sep 02 2022

Reducing the sequence modulo the odd integer 2*k + 1 results in a purely periodic sequence with period dividing 4*k + 2, For example, reduced modulo 5 the sequence becomes the purely periodic sequence [1, 3, 4, 2, 0, 4, 2, 1, 3, 0, 1, 3, 4, 2, 0, 4, 2, 1, 3, 0, ...] with period 10. - Peter Bala, Sep 12 2022

J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..200
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k
J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83. [Annotated scanned copy]
Simon Plouffe, Simple inverter lookup on 1.8134302039235
Simon Plouffe, Smart inverter lookup on 1.8134302039235
Index entries for sequences related to logarithmic numbers

Cf. A000166, A000522, A087208.

a:= proc(n) a(n):= n*`if`(n<2, n, (n-1)*a(n-2)-(-1)^n) end:
seq(a(n), n=1..25);  # Alois P. Heinz, Jul 10 2013

egf = x/Exp[x]/(1-x^2); a[n_] := SeriesCoefficient[egf, {x, 0, n}]*n!; Table[a[n], {n, 1, 22}] (* Jean-François Alcover, Jan 17 2014, after Vladeta Jovovic *)
a[n_] := (Exp[-1] Gamma[1 + n, -1] - (-1)^n Exp[1] Gamma[1 + n, 1])/2;
Table[a[n], {n, 1, 22}] (* Peter Luschny, Dec 18 2017 *)

a(n) = (-1)^(n+1)*sum(k=0, n, binomial(n, k)*k!*(1-(-1)^k)/2); \\ Michel Marcus, Jan 13 2022

E.g.f.: x/exp(x)/(1-x^2). - Vladeta Jovovic, Feb 09 2003
a(n) = n*((n-1)*a(n-2)-(-1)^n). - Matthew Vandermast, Jun 30 2003
From Gerald McGarvey, Jun 06 2004: (Start)
For n odd, a(n) = n! * Sum_{i=0..n-1, i even} 1/i!.
For n even, a(n) = n! * Sum_{i=1..n-1, i odd} 1/i!.
For n odd, lim_{n->infinity} a(n)/n! = cosh(1).
For n even, lim_{n->infinity} a(n)/n! = sinh(1).
For n even, lim_{n->infinity} n*a(n)*a(n-1)/n!^2 = cosh(1)*sinh(1).
For signed values, Sum_{n>=1} a(n)/n!^2 = 0.
For unsigned values, Sum_{n>=1} a(n)/n!^2 = cosh(1)*sinh(1). (End)
a(n) = (-1)^(n-1)*Sum_{k=0..n} C(n, k)*k!*(1-(-1)^k)/2. - Paul Barry, Sep 14 2004
a(n) = (-1)^(n+1)*n*A087208(n-1). - R. J. Mathar, Jul 24 2015
a(n) = (exp(-1)*Gamma(1+n,-1) - (-1)^n*exp(1)*Gamma(1+n,1))/2 = (A000166(n) - (-1)^n*A000522(n))/2. - Peter Luschny, Dec 18 2017

More terms from Jeffrey Shallit

More terms from Vladeta Jovovic, Feb 09 2003

A337749 a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k / (n-2*k)!.

Original entry on oeis.org

1, 1, -1, -5, 13, 101, -389, -4241, 21785, 305353, -1960649, -33588829, 258805669, 5239857325, -47102631757, -1100370038249, 11304631621681, 299300650403729, -3459217276234385, -102360822438075317, 1314502564969066301, 42991545423991633141, -607300185015708631061

Offset: 0

Ilya Gutkovskiy, Sep 18 2020

sign

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

Cf. A087208, A182386, A337750, A337751.

Table[n! Sum[(-1)^k/(n - 2 k)!, {k, 0, Floor[n/2]}], {n, 0, 22}]
nmax = 22; CoefficientList[Series[Exp[x]/(1 + x^2), {x, 0, nmax}], x] Range[0, nmax]!
(* alternative code *)
f[x_]:=I*(ExpIntegralE[-x,I]*E^I-ExpIntegralE[-x,-I]*E^(-I))/2
FunctionExpand[Array[f,20,0]] (* Velin Yanev, Oct 13 2021 *)

a(n) = n!*sum(k=0, n\2, (-1)^k / (n-2*k)!); \\ Michel Marcus, Sep 18 2020

G.f.: Sum_{k>=0} (-1)^k * (2*k)! * x^(2*k) / (1 - x)^(2*k+1).
E.g.f.: exp(x) / (1 + x^2).
a(0) = a(1) = 1; a(n) = 1 - n * (n-1) * a(n-2).

A291484 Expansion of e.g.f. arctanh(x)*exp(x).

Original entry on oeis.org

0, 1, 2, 5, 12, 49, 190, 1301, 7224, 69441, 495898, 6095429, 53005700, 792143793, 8110146070, 142633278997, 1679413757168, 33964965659649, 451969255722162, 10331348137881349, 153288815339260796, 3907452790559751857, 63949589015139119598, 1798373345567005989781, 32179694275204166066728

Offset: 0

Ilya Gutkovskiy, Aug 24 2017

nonn

			E.g.f.: A(x) = x/1! + 2*x^2/2! + 5*x^3/3! + 12*x^4/4! + 49*x^5/5! + ...

Cf. A002104, A002741, A009739, A009832, A010050, A012709, A087208 (first differences), A279927.

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

nmax = 24; Range[0, nmax]! CoefficientList[Series[ArcTanh[x] Exp[x], {x, 0, nmax}], x]
nmax = 24; Range[0, nmax]! CoefficientList[Series[Log[(1 + x)/(1 - x)] Exp[x]/2, {x, 0, nmax}], x]
nmax = 24; Range[0, nmax]! CoefficientList[Series[Sum[x^(2 k + 1)/(2 k + 1), {k, 0, Infinity}] Exp[x], {x, 0, nmax}], x]
Table[Sum[Binomial[n+1,2k+1](n-2k)/(n+1) (2 k)!, {k,0,n/2}],{n,0,12}] (* Emanuele Munarini, Dec 16 2017 *)

makelist(sum(binomial(n+1,2*k+1)*(n-2*k)/(n+1)*(2*k)!,k,0,floor(n/2)),n,0,12); /* Emanuele Munarini, Dec 16 2017 */

first(n) = x='x+O('x^n); Vec(serlaplace(atanh(x)*exp(x)), -n) \\ Iain Fox, Dec 16 2017

E.g.f.: log((1 + x)/(1 - x))*exp(x)/2.
From Emanuele Munarini, Dec 16 2017: (Start)
a(n) = Sum_{k=0..n/2} binomial(n+1,2*k+1)*((n-2*k)/(n+1))*(2*k)!.
a(n+3) - a(n+2) - (n+1)*(n+2)*a(n+1) + (n+1)*(n+2)*a(n) = 1.
a(n+4) - 2*a(n+3) - (n^2+5*n+5)*a(n+2) + 2*(n+2)^2*a(n+1) - (n+1)*(n+2)*a(n) = 0.
(End)
a(n) ~ (n-1)! * (exp(1) - (-1)^n * exp(-1))/2. - Vaclav Kotesovec, Dec 16 2017

A334157 Row sums of array A158777.

Original entry on oeis.org

1, 2, 5, 16, 89, 686, 5917, 54860, 588401, 7370074, 103522421, 1573237832, 25869057865, 462768222086, 8965777751309, 186025937645956, 4106375449878497, 96241703493486770, 2390797380938894821, 62730027061416412544

Offset: 0

Petros Hadjicostas, Apr 16 2020

nonn

Cf. A003269, A087208, A158777, A330044, A330045.

W := proc(n, m) local v, s, h; v := 0;
for s from 0 to m do
if 0 = (m - s) mod 4 then
h := (m - s)/4;
v := v + binomial(n - s - 3*h, h)/s!;
end if; end do; n!*v; end proc;
seq(add(W(n1, m1), m1 = 0 .. n1), n1 = 0 .. 35);

Table[Apply[Plus, CoefficientList[Expand[t^n*n!*SeriesCoefficient[Series[Exp[t*x]/( 1 - x/t - t^4*x^4), {x, 0, 50}], n]], t]], {n, 0, 40}]; (* Program due to Roger L. Bagula from A158777 *)

a(n) = n!*Sum_{k=0..n} A003269(k+1)/(n-k)!.
a(n) = n!*Sum_{k=0..n} Sum_{s=0..floor(k/3)} binomial(k-3*s, s)/(n-k)!.
E.g.f.: exp(x)/(1 - x - x^4).

A114633 a(n) = (n+1)(n+2)/2 Sum_{k=0..floor(n/2)} n!/(n-2*k)!.

Original entry on oeis.org

1, 3, 18, 70, 555, 2961, 31108, 213228, 2799765, 23455135, 369569046, 3659001138, 67261566463, 768390239085, 16142775951240, 209002145031256, 4939689441079593, 71478733600689723, 1877081987610245530, 30021068112289683870, 867211878275933435091, 15190660464818580038473

Offset: 0

Creighton Dement, Feb 17 2006

nonn

Formula was found by Paul D. Hanna.

Related to logarithmic numbers A002104.

Vincenzo Librandi, Table of n, a(n) for n = 0..450

Cf. A002104, A087208.

a:= n-> (n+1)*(n+2)/2*sum(n!/(n-2*k)!,k=0..floor(n/2)): seq(a(n), n=0..20);

Rest[Rest[With[{nn=25}, CoefficientList[Series[Exp[x]/(1 - x^2)(x^2/2), {x, 0, nn}], x] Range[0, nn]!]]] (* Vincenzo Librandi, Sep 03 2017 *)

a(n) = A087208(n)*(n+1)*(n+2)/2. - Paul D. Hanna

E.g.f.: exp(x)/(1-x^2)*(x^2/2) (with offset 2). - Zerinvary Lajos, Apr 03 2009

More terms from Vincenzo Librandi, Sep 03 2017

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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A330044 Expansion of e.g.f. exp(x) / (1 - x^3).

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A330045 Expansion of e.g.f. exp(x) / (1 - x^4).

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A002747 Logarithmic numbers.

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A337749 a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k / (n-2*k)!.

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A291484 Expansion of e.g.f. arctanh(x)*exp(x).

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

A114633 a(n) = (n+1)(n+2)/2 Sum_{k=0..floor(n/2)} n!/(n-2*k)!.