A089148 -id:A089148 - cp's OEIS frontend

A006153 E.g.f.: 1/(1-x*exp(x)).

1, 1, 4, 21, 148, 1305, 13806, 170401, 2403640, 38143377, 672552730, 13044463641, 276003553860, 6326524990825, 156171026562838, 4130464801497105, 116526877671782896, 3492868475952497313, 110856698175372359346, 3713836169709782989993, 130966414749485504586940

Offset: 0

Simon Plouffe and N. J. A. Sloane

a(n) is the sum of the row entries of triangle A199673, that is, a(n) is the number of ways to assign n people into labeled groups and then to assign a leader for each group from its members; see example below. - Dennis P. Walsh, Nov 15 2011
a(n) is the number of functions f:{1,2,...,n}->{1,2,...,n} (endofunctions) such that for some j>1, f^j=f where f^j denotes iterated functional composition. Equivalently, the number of endofunctions such that every element is mapped to a recurrent element. Equivalently, every vertex of the functional digraph is at a distance at most 1 from a cycle. - Geoffrey Critzer, Jan 21 2012
Numerators in rational approximations of Lambert W(1). See Ramanujan, Notebooks, volume 2, page 22: "2. If e^{-x} = x, shew that the convergents to x are 1/2, 4/7, 21/37, 148/261, &c." - Michael Somos, Jan 21 2019

			a(3) = 21 since there are 21 ways to assign 3 people into labeled groups with designated leaders. If there is one group, there are 3 ways to select a leader from the 3 people in the group. If there are two groups (group 1 and group 2), there are 6 ways to assign leaders and then 2 ways to select a group for the remaining person, and thus there are 12 assignments. If there are three groups (group1, group 2, and group3), each person is a leader of their singleton group, and there are 6 ways to assign the 3 people to the 3 groups. Hence a(3) = 3 + 12 + 6 = 21.
a(4) = 148 = 4 + 48 + 72 + 24.

S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 2, see page 22.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.32(d).

Alois P. Heinz, Table of n, a(n) for n = 0..200
S. Getu and L. W. Shapiro, Combinatorial view of the composition of functions, Ars Combin. 10 (1980), 131-145. (Annotated scanned copy)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 110
Dennis Walsh, Assigning people into labeled groups with leaders

Row sums of triangle A199673.

Cf. A003566, A072597, A089148.

a := proc(n) local k; add(k^(n-k)*n!/(n-k)!,k=1..n); end; # for n >= 1

With[{nn=20},CoefficientList[Series[1/(1-x Exp[x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 29 2012 *)
a[ n_] := If[n < 0, 0, n! + n! Sum[(n - k)^k / k!, {k, n}]]; (* Michael Somos, Jan 21 2019 *)

x='x+O('x^66);
egf=1/(1-x*exp(x)); /* = 1 + x + 2*x^2 + 7/2*x^3 + 37/6*x^4 + 87/8*x^5 +... */
Vec(serlaplace(egf)) /* Joerg Arndt, Apr 30 2011 */

{a(n) = if(n<0, 0, n! * sum(k=0, n, (n-k)^k / k!))}; /* Michael Somos, Jan 21 2019 */

def A006153_list(len):
    f, R, C = 1, [1], [1]+[0]*(len-1)
    for n in (1..len-1):
        f *= n
        for k in range(n, 0, -1):
            C[k] = -C[k-1]*(1/(k-1) if k>1 else 1)
        C[0] = sum((-1)^k*C[k] for k in (1..n))
        R.append(C[0]*f)
    return R
print(A006153_list(20)) # Peter Luschny, Feb 21 2016

a(n) = n! * Sum_{k=0..n}(n-k)^k/k!.
a(n) = Sum_{k=0..n} k!*k^(n-k)*binomial(n,k).
For n>=1, a(n-1) = b(n) where b(1)=1 and b(n) = Sum_{i=1..n-1} i*binomial(n-1, i)*b(i). - Benoit Cloitre, Nov 13 2004
a(n) = Sum_{k=1..n}A199673(n,k) = Sum_{k=1..n}n! k^(n-k)/(n-k)!. - Dennis P. Walsh, Nov 15 2011
E.g.f. for a(n), n>=1: x*e^x/(1-x*e^x). - Dennis P. Walsh, Nov 15 2011
a(n) ~ n! / ((1+LambertW(1))*LambertW(1)^n). - Vaclav Kotesovec, Jun 21 2013
O.g.f.: Sum_{n>=0} n! * x^n / (1 - n*x)^(n+1). - Paul D. Hanna, May 22 2018
a(0) = 1; a(n) = n * Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Ilya Gutkovskiy, Jul 12 2020

Definition corrected by Joerg Arndt, Apr 30 2011

A072597 Expansion of 1/(exp(-x) - x) as exponential generating function.

Original entry on oeis.org

1, 2, 7, 37, 261, 2301, 24343, 300455, 4238153, 67255273, 1185860331, 23000296155, 486655768525, 11155073325917, 275364320099807, 7282929854486431, 205462851526617489, 6158705454187353297, 195465061563672788947, 6548320737474275229347, 230922973019493881984021

Offset: 0

Michael Somos, Jun 23 2002

Polynomials from A140749/A141412 are linked to Stirling1 (see A048594, A129841, A140749). See also P. Flajolet, X. Gourdon, B. Salvy in, available on Internet, RR-1857.pdf (preprint of unavailable Gazette des Mathematiciens 55, 1993, pp. 67-78; for graph 2 see also X. Gourdon RR-1852.pdf, pp. 64-65). What is the corresponding graph for A152650/A152656 = simplified A009998/A119502 linked, via A152818, to a(n), then Stirling2? - Paul Curtz, Dec 16 2008
Denominators in rational approximations of Lambert W(1). See Ramanujan, Notebooks, volume 2, page 22: "2. If e^{-x} = x, shew that the convergents to x are 1/2, 4/7, 21/37, 148/261, &c." Numerators in A006153. - Michael Somos, Jan 21 2019
Call an element g in a semigroup a group element if g^j = g for some j > 1. Then a(n) is the number of group elements in the semigroup of partial transformations of an n-set. Hence a(n) = Sum_{k=0..n} A154372(n,k)*k!. - Geoffrey Critzer, Nov 27 2021

			G.f. = 1 + 2*x + 7*x^2 + 37*x^3 + 261*x^4 + 2301*x^5 + 24343*x^6 + ...

O. Ganyushkin and V Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009, page 70.
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 2, see page 22.

Seiichi Manyama, Table of n, a(n) for n = 0..411
W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013.
G. Jiraskova and J. Shallit, The state complexity of star-complement-star, arXiv preprint arXiv:1203.5353 [cs.FL], 2012. - From _N. J. A. Sloane_, Sep 21 2012

Cf. A000110, A006153, A030178, A089148.
Cf. A048993, A052820.
Cf. A006153, A154372.

CoefficientList[Series[1/(Exp[-x]-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 26 2013 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ 1 / (Exp[-x] - x), {x, 0, n}]]; (* Michael Somos, Jan 21 2019 *)
a[ n_] := If[ n < 0, 0, n! Sum[ (n - k + 1)^k / k!, {k, 0, n}]]; (* Michael Somos, Jan 21 2019 *)

{a(n) = if( n<0, 0, n! * polcoeff( 1 / (exp(-x + x * O(x^n)) - x), n))};

{a(n) = if( n<0, 0, n! * sum(k=0, n, (n-k+1)^k / k!))}; /* Michael Somos, Jan 21 2019 */

E.g.f.: 1 / (exp(-x) - x).
a(n) = n!*Sum_{k=0..n} (n-k+1)^k/k!. - Vladeta Jovovic, Aug 31 2003
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*A052820(k). - Vladeta Jovovic, Apr 12 2004
Recurrence: a(n+1) = 1 + Sum_{j=1..n} binomial(n, j)*a(j)*j. - Jon Perry, Apr 25 2005
E.g.f.: 1/(Q(0) - x) where Q(k) = 1 - x/(2*k+1 - x*(2*k+1)/(x - (2*k+2)/Q(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Apr 04 2013
a(n) ~ n!/((1+c)*c^(n+1)), where c = A030178 = LambertW(1) = 0.5671432904... - Vaclav Kotesovec, Jun 26 2013
O.g.f.: Sum_{k>=0} k!*x^k/(1 - (k + 1)*x)^(k+1). - Ilya Gutkovskiy, Oct 09 2018
a(n) = A006153(n+1)/(n+1). - Seiichi Manyama, Nov 05 2024

A302397 Expansion of e.g.f. 1/(1 + x*exp(x)).

Original entry on oeis.org

1, -1, 0, 3, -4, -25, 114, 287, -4152, 1647, 192230, -807961, -10164804, 111209111, 454840554, -14657978385, 21202175504, 1988791958879, -15488971798194, -260886468394153, 4872247004699460, 23537372210149959, -1365745577227898350, 4274609859520565663, 364461939727273277016

Offset: 0

Ilya Gutkovskiy, Apr 07 2018

sign

			1/(1 + x*exp(x)) = 1 - x/1! + 3*x^3/3! - 4*x^4/4! - 25*x^5/5! + 114*x^6/6! + 287*x^7/7! - 4152*x^8/8! + 1647*x^9/9! + ...

Seiichi Manyama, Table of n, a(n) for n = 0..474

Cf. A006153, A009306, A089148.

a:=series(1/(1+x*exp(x)),x=0,25): seq(n!*coeff(a,x,n),n=0..24); # Paolo P. Lava, Mar 26 2019

nmax = 24; CoefficientList[Series[1/(1 + x Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[n! Sum[(-1)^(n - k) (n - k)^k/k!, {k, 0, n}], {n, 24}]]
Join[{1}, Table[Sum[(-1)^k k! k^(n - k) Binomial[n, k], {k, 0, n}], {n, 24}]]

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

A152818 Array read by antidiagonals: A(n,k) = (k+1)^n*(n+k)!/n!.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 12, 18, 6, 1, 32, 108, 96, 24, 1, 80, 540, 960, 600, 120, 1, 192, 2430, 7680, 9000, 4320, 720, 1, 448, 10206, 53760, 105000, 90720, 35280, 5040, 1, 1024, 40824, 344064, 1050000, 1451520, 987840, 322560, 40320

Offset: 0

Paul Curtz, Dec 13 2008

A009998/A119502 gives triangle of unreduced coefficients of polynomials defined by A152650/A152656. a(n) gives numerators with denominators n! for each row.

Row 0 is A000142. Row 1 is formed from positive members of A001563. Row 2 is A055533. Column 0 is A000012. Column 1 is formed from positive members of A001787. Column 2 is A006043. Column 3 is A006044. - Omar E. Pol, Jan 06 2009

			From _Omar E. Pol_, Jan 06 2009: (Start)
Array begins:
  1,    1,      2,        6,         24,          120, ...
  1,    4,     18,       96,        600,         4320, ...
  1,   12,    108,      960,       9000,        90720, ...
  1,   32,    540,     7680,     105000,      1451520, ...
  1,   80,   2430,    53760,    1050000,     19595520, ...
  1,  192,  10206,   344064,    9450000,    235146240, ...
  1,  448,  40824,  2064384,   78750000,   2586608640, ...
  1, 1024, 157464, 11796480,  618750000,  26605117440, ...
  1, 2304, 590490, 64880640, 4640625000, 259399895040, ... (End)
Antidiagonal triangle:
  1;
  1,   1;
  1,   4,     2;
  1,  12,    18,     6;
  1,  32,   108,    96,     24;
  1,  80,   540,   960,    600,   120;
  1, 192,  2430,  7680,   9000,  4320,   720;
  1, 448, 10206, 53760, 105000, 90720, 35280, 5040;

G. C. Greubel, Antidiagonals n = 0..50, flattened
F. A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424. See page 422.
F. A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424. (Annotated scanned copy)
F. A. Haight, Letter to N. J. A. Sloane, n.d.

Cf. A000012, A000142, A001563, A001787, A006043, A006044, A009998.
Cf. A055533, A072597 (antidiagonal sums), A089148, A119502, A152650.
Cf. A152656, A154372.

A152818:= func< n,k | (k+1)^(n-k)*Factorial(k)*Binomial(n,k) >;
[A152818(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 10 2023

len= 45; m= 1 + Ceiling[Sqrt[len]]; Sort[Flatten[#, 1] &[MapIndexed[ {(2 +#2[[1]]^2 +(#2[[2]] -1)*#2[[2]] +#2[[1]]*(2*#2[[2]] -3))/ 2, #1}&, Table[(k+1)^n*(n+k)!/n!, {n,0,m}, {k,0,m}], {2}]]][[All, 2]][[1 ;; len]] (* From Jean-François Alcover, May 27 2011 *)
T[n_, k_]:= (k+1)^(n-k)*k!*Binomial[n, k];
Table[T[n,k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 10 2023 *)

A(n,k) = (k+1)^n*(n+k)!/n! \\ Charles R Greathouse IV, Sep 10 2016

def A152818_row(n):
    R. = ZZ[]
    P = add((n-k+1)^k*x^(n-k+1)*factorial(n)/factorial(k) for k in (0..n))
    return P.coefficients()
for n in (0..12): print(A152818_row(n))  # Peter Luschny, May 03 2013

E.g.f. for array as a triangle: exp(x)/(1-t*x*exp(x)) = 1+(1+t)*x+(1+4*t+2*t^2)*x^2/2! + (1+12*t+18*t^2+6*t^3)*x^3/3! + .... E.g.f. is int {z = 0..inf} exp(-z)*F(x,t*z), (x and t chosen sufficiently small for the integral to converge), where F(x,t) = exp(x*(1+t*exp(x))) is the e.g.f. for A154372. - Peter Bala, Oct 09 2011
From Peter Bala, Oct 09 2011: (Start)
From the e.g.f., the row polynomials R(n,t) satisfy the recursion R(n,t) = 1 + t*sum {k = 0..n-1} n!/(k!*(n-k-1)!)*R(n-k-1,t). The polynomials 1/n!*R(n,x) are the polynomials P(n,x) of A152650.
Sum_{k=0..n} T(n, k) = A072597(n) (antidiagonal sums). (End)
From G. C. Greubel, Apr 10 2023: (Start)
T(n, k) = (k+1)^(n-k) * k! * binomial(n, k) (antidiagonal triangle).
Sum_{k=0..n} (-1)^k*T(n, k) = A089148(n). (End)

Better definition, extended and edited by Omar E. Pol and N. J. A. Sloane, Jan 05 2009

A305990 Expansion of e.g.f.: (1+x) / (exp(-x) - x).

Original entry on oeis.org

1, 3, 11, 58, 409, 3606, 38149, 470856, 6641793, 105398650, 1858413061, 36044759796, 762659322385, 17481598316742, 431535346662645, 11413394655983536, 321989729198400385, 9651573930139850610, 306321759739045148293, 10262156907184058219340

Offset: 0

Vaclav Kotesovec, Jun 16 2018

nonn

Cf. A305133, A006153, A009306, A009444, A072597, A089148, A302397.

nmax = 20; CoefficientList[Series[(1+x)/(E^(-x)-x), {x, 0, nmax}], x] * Range[0, nmax]!
a={1};For[n=1,n<20,n++,AppendTo[a,Sum[(n!)*((n-k+1)^(k-1))*(n+1)/(k!),{k,0,n+1}]]]; a (* Detlef Meya, Sep 05 2023 *)

a(n) ~ n! / LambertW(1)^(n+1).
a(n) = (-1)^n * A009444(n+1).
a(n) = Sum_{k=0..n+1} (n+1)!*(n-k+1)^(k-1)/k! for n > 0. - Detlef Meya, Sep 05 2023

A352303 Expansion of e.g.f. 1/(exp(x) - x^3).

Original entry on oeis.org

1, -1, 1, 5, -47, 239, -239, -11761, 170689, -1237825, -2360159, 238756319, -4146035519, 32586126143, 359988680689, -18567245926321, 351652342984321, -2283764958280321, -89760640709677247, 3866819337993369023, -74731210747948586879, 167887841949213912959

Offset: 0

Seiichi Manyama, Mar 11 2022

sign

Cf. A089148, A352302, A352304.

Cf. A352299.

m = 21; Range[0, m]! * CoefficientList[Series[1/(Exp[x] - x^3), {x, 0, m}], x] (* Amiram Eldar, Mar 12 2022 *)

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(exp(x)-x^3)))

b(n, m) = if(n==0, 1, sum(k=1, n, (-1+(k==m)*m!)*binomial(n, k)*b(n-k, m)));
a(n) = b(n, 3);

a(n) = n * (n-1) * (n-2) * a(n-3) - Sum_{k=1..n} binomial(n,k) * a(n-k) for n > 2.

a(n) = n! * Sum_{k=0..floor(n/3)} (-k-1)^(n-3*k)/(n-3*k)!. - Seiichi Manyama, Aug 21 2024

A266328 E.g.f. A(x) satisfies: A(x) = exp( Integral B(x) dx ) such that B(x) = exp(-x) * exp( Integral A(x) dx ), where the constant of integration is zero.

Original entry on oeis.org

1, 1, 1, 2, 6, 21, 92, 469, 2731, 17985, 131528, 1059616, 9319363, 88833422, 912393381, 10043727089, 117969438513, 1472593659884, 19467505081458, 271704942613323, 3992343851680466, 61603531051030691, 995949139457447931, 16835191741257445589, 296976010796327785530, 5457427389713208932740, 104308245862443706265341, 2070461793105333579698992, 42622090166454492404075635

Offset: 0

Paul D. Hanna, Jan 24 2016

nonn

Compare to: G(x) = exp( Integral G(x) dx ) when G(x) = 1/(1-x).
What is lim_{n->oo} (a(n)/n!)^(1/n)?  Example: (a(500)/500!)^(1/500) = 0.7353325805...
Limit_{n->oo} (a(n)/n!)^(1/n) = 1/Integral_{x=0..oo} 1/(exp(x) - x) dx = 0.73578196429164719984313538... - Vaclav Kotesovec, Aug 21 2017

			E.g.f.: A(x) = 1 + x + x^2/2! + 2*x^3/3! + 6*x^4/4! + 21*x^5/5! + 92*x^6/6! + 469*x^7/7! + 2731*x^8/8! + 17985*x^9/9! + 131528*x^10/10! + ...
such that log(A(x)) = Integral B(x) dx
where
B(x) = 1 + x^2/2! + x^3/3! + 5*x^4/4! + 16*x^5/5! + 76*x^6/6! + 393*x^7/7! + 2338*x^8/8! + 15647*x^9/9! + 115881*x^10/10! + ...
and A(x) and B(x) satisfy:
(1) A(x) = B'(x)/B(x) + 1,
(2) B(x) = A'(x)/A(x),
(3) B(x) = A(x) - log(A(x)),
(4) log(A(x)) = Integral B(x) dx,
(5) log(B(x)) = Integral A(x) dx - x.
The Series Reversion of log(A(x)) equals Integral 1/(exp(x) - x) dx:
Integral 1/(exp(x) - x) dx  =  x - x^3/3! - x^4/4! + 5*x^5/5! + 19*x^6/6! - 41*x^7/7! - 519*x^8/8! - 183*x^9/9! + 19223*x^10/10! + ... + A089148(n-1)*x^n/n! + ...
so that A( Integral 1/(exp(x) - x) dx ) = exp(x).

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A006789, A089148, A266329.

{a(n) = my(A=1+x,B=1+x); for(i=0,n, A = exp( intformal( B + x*O(x^n) ) ); B = exp( intformal( A - 1 ) ) ); n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

{a(n) = n! * polcoeff( exp( serreverse( intformal( 1/(exp(x +x*O(x^n)) - x) ) )), n)}
for(n=0,30,print1(a(n),", "))

Vec( serlaplace( exp( serreverse( intformal( 1/(exp(x +x*O(x^25)) - x)))))) \\ Joerg Arndt, Dec 26 2023

E.g.f. A(x) satisfies:
(1) A(x) = exp( Integral A(x) - log(A(x)) dx ).
(2) A(x) = log(A(x)) + A'(x)/A(x).
(3) log(A(x)) = exp(-x) * Integral exp(x)*A(x) dx.
(4) A(x) = exp( Series_Reversion( Integral 1/(exp(x) - x) dx ) ).
a(n) ~ c^(n+1) * n!, where c = 1/Integral_{x=0..oo} 1/(exp(x) - x) dx = 0.7357819642916471998431353808137704665788888148929882090175... - Vaclav Kotesovec, Aug 21 2017
Conjecture: a(n) = R(n-1, 0) for n > 0 with a(0) = 1 where R(n, q) = R(n-1, q+1) + Sum_{j=0..q-1} binomial(q+1, j)*R(n-1, j) for n > 0, q >= 0 with R(0, q) = 1 for q >= 0. - Mikhail Kurkov, Dec 26 2023

A352304 Expansion of e.g.f. 1/(exp(x) - x^4).

Original entry on oeis.org

1, -1, 1, -1, 25, -241, 1441, -6721, 67201, -1185409, 16652161, -180639361, 2098673281, -37526586241, 785718950017, -14516030954881, 247504017895681, -4832929862019841, 116556246644716801, -2930255897793414913, 69746855593499124481, -1673960044278244020481

Offset: 0

Seiichi Manyama, Mar 11 2022

sign

Cf. A089148, A352302, A352303.

Cf. A352300, A352311.

m = 21; Range[0, m]! * CoefficientList[Series[1/(Exp[x] - x^4), {x, 0, m}], x] (* Amiram Eldar, Mar 12 2022 *)

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(exp(x)-x^4)))

b(n, m) = if(n==0, 1, sum(k=1, n, (-1+(k==m)*m!)*binomial(n, k)*b(n-k, m)));
a(n) = b(n, 4);

a(n) = n * (n-1) * (n-2) * (n-3) * a(n-4) - Sum_{k=1..n} binomial(n,k) * a(n-k) for n > 3.
a(n) ~ n! * (-1)^n / ((1 + LambertW(1/4)) * 2^(2*n + 10) * LambertW(1/4)^(n+4)). - Vaclav Kotesovec, Mar 12 2022
a(n) = n! * Sum_{k=0..floor(n/4)} (-k-1)^(n-4*k)/(n-4*k)!. - Seiichi Manyama, Aug 21 2024

A336958 E.g.f.: 1 / (exp(2*x) - x).

Original entry on oeis.org

1, -1, -2, 10, 24, -312, -560, 19472, 6272, -1994624, 4072704, 299059968, -1635814400, -60723321856, 628215191552, 15716076562432, -274420622327808, -4900668238036992, 140182198527655936, 1717697481518809088, -83651335147070685184, -590374211868638314496

Offset: 0

Ilya Gutkovskiy, Aug 09 2020

sign

Cf. A089148, A336947, A336959.

nmax = 21; CoefficientList[Series[1/(Exp[2 x] - x), {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-2 (n - k + 1))^k/k!, {k, 0, n}], {n, 0, 21}]
a[0] = 1; a[n_] := a[n] = -n a[n - 1] - Sum[Binomial[n, k] 2^k a[n - k], {k, 2, n}]; Table[a[n], {n, 0, 21}]

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

a(0) = 1; a(n) = -n * a(n-1) - Sum_{k=2..n} binomial(n,k) * 2^k * a(n-k).

A352302 Expansion of e.g.f. 1/(exp(x) - x^2).

Original entry on oeis.org

1, -1, 3, -13, 73, -521, 4441, -44185, 502545, -6429169, 91393201, -1429101521, 24378097129, -450504733849, 8965682806809, -191174795868841, 4348171177591201, -105077942935229537, 2688685949077138657, -72618903735812907553, 2064598911185525708601

Offset: 0

Seiichi Manyama, Mar 11 2022

sign

Cf. A089148, A352303, A352304.

Cf. A346269.

m = 20; Range[0, m]! * CoefficientList[Series[1/(Exp[x] - x^2), {x, 0, m}], x] (* Amiram Eldar, Mar 12 2022 *)

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(exp(x)-x^2)))

b(n, m) = if(n==0, 1, sum(k=1, n, (-1+(k==m)*m!)*binomial(n, k)*b(n-k, m)));
a(n) = b(n, 2);

a(n) = n * (n-1) * a(n-2) - Sum_{k=1..n} binomial(n,k) * a(n-k) for n > 1.
a(n) ~ n! * (-1)^n / ((1 + LambertW(1/2)) * 2^(n+3) * LambertW(1/2)^(n+2)). - Vaclav Kotesovec, Mar 12 2022
a(n) = n! * Sum_{k=0..floor(n/2)} (-k-1)^(n-2*k)/(n-2*k)!. - Seiichi Manyama, Aug 21 2024

cp's OEIS Frontend

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