A052845 -id:A052845 - cp's OEIS frontend

A088009 Number of "sets of odd lists", cf. A000262.

1, 1, 1, 7, 25, 181, 1201, 10291, 97777, 1013545, 12202561, 151573951, 2173233481, 31758579997, 524057015665, 8838296029291, 164416415570401, 3145357419120721, 65057767274601217, 1391243470549894135, 31671795881695430521, 747996624368605997701

Offset: 0

Vladeta Jovovic, Nov 02 2003

The Brauer algebra has a basis consisting of all graphs on the vertex set {1,...,2n} whose vertices all have degree 1. The multiplication is defined in Halverson and Ram. a(n) is also the number of idempotent basis elements (i.e., those satisfying b^2=b) of the Brauer algebra. - James East, Dec 27 2013
From Peter Bala, Nov 26 2017: (Start)
The sequence terms have the form 6*m + 1 (follows from the recurrence).
a(n+k) = a(n) (mod k) for all n and k. It follows that the sequence a(n) (mod k) is periodic with the exact period dividing k. For example, modulo 10 the sequence becomes 1, 1, 1, 7, 5, 1, 1, 1, 7, 5, ... with exact period 5. (End)

			From _R. J. Mathar_, Feb 01 2022 (Start):
Examples of partitions of elements {1,2,..n} into sets of lists where each list contains an odd number of elements:
n=1: One set where the element is the list.
n=2: One set where each of the 2 elements is its own list.
n=3: One set where each of the 3 elements is its own list, plus 6=3! sets of a list of all 3 elements.
n=4: One set where each of the 4 elements is its own list, plus 4*3! sets where one (4 choices) element is its own list and the remaining 3 elements are in another list.
n=5: One set where each of the 5 elements is its own list, plus 5!=120 sets where all 5 elements are in the same list, plus binomial(5,2)*3!=60 sets where two elements are in their own lists and the other 3 in a third list. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..446 (terms 0..200 from Alois P. Heinz)
I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, et al., Enumeration of idempotents in diagram semigroups and algebras, arXiv preprint arXiv:1408.2021 [math.GR], 2014.
T. Halverson and A. Ram, Partition algebras, arXiv:math/0401314 [math.RT], 2004.
T. Halverson and A. Ram, Partition algebras, European J. Combin. 26 (6) (2005) 869-921.

Cf. A052845, A088026, A000010, A000262, A168561, A318976.

T:= (n, k)-> `if`(n-k mod 2 = 0, binomial((n+k)/2, k), 0):
a:= n-> n! * add(T(n-1, k-1)/k!, k=0..n):
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 07 2011
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add((i->
      a(n-i)*binomial(n-1, i-1)*i!)(2*j+1), j=0..(n-1)/2))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Feb 01 2022

a[n_] := SeriesCoefficient[ Exp[x/(1 - x^2) ], {x, 0, n}]*n!; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 24 2015 *)

x='x+O('x^33);
Vec(serlaplace(exp(x/(1-x^2))))
/* Joerg Arndt, Mar 09 2011 */

E.g.f.: exp(x/(1-x^2)).
a(n) = n!*Sum_{k=1..n} A168561(n-1,k-1)/k!. - Vladimir Kruchinin, Mar 07 2011
E.g.f.: 1 + x/(G(0)-x) where G(k)= (1-x^2)*k + 1+x-x^2 - x*(1-x^2)*(k+1)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Aug 02 2012
E.g.f.: 1 + x/(1+x)*(G(0) - 1) where G(k) = 1 + 1/(1+x^2)/(k+1)/(1-x/(x+(1)/G(k+1))), (continued fraction). - Sergei N. Gladkovskii, Feb 04 2013
a(n) ~ 2^(-3/4)*n^(n-1/4)*exp(sqrt(2*n)-n) * (1-11/(24*sqrt(2*n))). - Vaclav Kotesovec, Aug 10 2013
D-finite with recurrence a(n) = a(n-1) + 2*(n-2)*(n-1)*a(n-2) + (n-2)*(n-1)*a(n-3) - (n-4)*(n-3)*(n-2)*(n-1)*a(n-4). - Vaclav Kotesovec, Aug 10 2013
E.g.f.: Product_{n >= 1} (1 + x^n)^(phi(n)/n) = Product_{n >= 0} ( (1 + x^(2*n+1))/(1 - x^(2*n+1)) )^( phi(2*n+1)/(4*n + 2) ), where phi(n) = A000010(n) is the Euler totient function. Cf. A066668 and A000262. - Peter Bala, Jan 01 2014
E.g.f.: Product_{k>0} exp(x^(2*k-1)). - Seiichi Manyama, Oct 10 2017

Prepended a(0)=1 by Joerg Arndt, Jul 29 2012

A113235 Number of partitions of {1,..,n} into any number of lists of size not equal to 2, where a list means an ordered subset, cf. A000262.

Original entry on oeis.org

1, 1, 1, 7, 49, 301, 2281, 21211, 220417, 2528569, 32014801, 442974511, 6638604721, 107089487077, 1849731389689, 34051409587651, 665366551059841, 13751213558077681, 299644435399909537, 6864906328749052759, 164941239260973870001, 4146673091958686331421

Offset: 0

Karol A. Penson, Oct 19 2005

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..444

Cf. A000262, A000266, A005251, A052845, A097514.

This sequence, A113236 and A113237 all describe the same type of mathematical structure: lists with some restrictions.

I:=[1, 1, 7, 49]; [1] cat [n le 4 select I[n] else (2*n-1)*Self(n -1) - (n-1)*n*Self(n-2) +4*(n-1)*(n-2)*Self(n-3) -2*(n-1)*(n-2)*(n-3)* Self(n-4): n in [1..30]]; // G. C. Greubel, May 16 2018

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*j!, j=[1, $3..n]))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, May 10 2016

f[n_] := n!*Sum[(-1)^k*LaguerreL[n - 2*k, -1, -1]/k!, {k, 0, Floor[n/2]}]; Table[ f[n], {n, 0, 19}]
Range[0, 19]!*CoefficientList[ Series[ Exp[x*(1 - x + x^2)/(1 - x)], {x, 0, 19}], x] (* Robert G. Wilson v, Oct 21 2005 *)

m=30; v=concat([1,1,7,49], vector(m-4)); for(n=5, m, v[n]=(2*n-1)*v[n-1]-(n-1)*n*v[n-2]+4*(n-1)*(n-2)*v[n-3]-2*(n-1)*(n-2)*(n-3)*v[n -4]); concat([1], v) \\ G. C. Greubel, May 16 2018

x='x+O('x^99); Vec(serlaplace(exp(x*(1-x+x^2)/(1-x)))) \\ Altug Alkan, May 17 2018

Expression as a sum involving generalized Laguerre polynomials, in Mathematica notation: a(n)=n!*Sum[(-1)^k*LaguerreL[n - 2*k, -1, -1]/k!, {k, 0, Floor[n/2]}], n=0, 1... .
E.g.f.: exp(x*(1-x+x^2)/(1-x)).
From Vaclav Kotesovec, Nov 13 2017: (Start)
a(n) = (2*n - 1)*a(n-1) - (n-1)*n*a(n-2) + 4*(n-2)*(n-1)*a(n-3) - 2*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ exp(-3/2 + 2*sqrt(n) - n) * n^(n-1/4) / sqrt(2) * (1 + 91/(48*sqrt(n))).
(End)

A129652 Exponential Riordan array [e^(x/(1-x)),x].

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 13, 9, 3, 1, 73, 52, 18, 4, 1, 501, 365, 130, 30, 5, 1, 4051, 3006, 1095, 260, 45, 6, 1, 37633, 28357, 10521, 2555, 455, 63, 7, 1, 394353, 301064, 113428, 28056, 5110, 728, 84, 8, 1, 4596553, 3549177, 1354788, 340284, 63126, 9198, 1092, 108, 9, 1

Offset: 0

Paul Barry, Apr 26 2007

Satisfies the equation e^[x/(1-x),x] = e*[e^(x/(1-x)),x].
Row sums are A052844.
Antidiagonal sums are A129653.

			Triangle begins:
     1;
     1,    1;
     3,    2,    1;
    13,    9,    3,   1;
    73,   52,   18,   4,  1;
   501,  365,  130,  30,  5, 1;
  4051, 3006, 1095, 260, 45, 6, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
T.-X. He, A symbolic operator approach to power series transformation-expansion formulas, JIS 11 (2008) 08.2.7.

Cf. A000262 (column 0), A052844 (row sums).
T(2n,n) gives A350461.
Cf. A052845, A111884, A133314.

A129652 := (n, k) -> (-1)^(k-n+1)*binomial(n,k)*KummerU(k-n+1, 2, -1);
seq(seq(round(evalf(A129652(n,k),99)),k=0..n),n=0..9); # Peter Luschny, Sep 17 2014
# second Maple program:
b:= proc(n) option remember; `if`(n=0, [1$2], add((p-> p+
     [0, p[1]*x^j])(b(n-j)*binomial(n-1, j-1)*j!), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i)/i!, i=0..n))(b(n)[2]):
seq(T(n), n=0..10);  # Alois P. Heinz, Feb 21 2022

T[n_, k_] := If[k==n, 1, n!/k! Sum[Binomial[n-k-1, j]/(j+1)!, {j, 0, n-k-1}]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Jean-François Alcover, Jun 14 2019 *)

Number triangle T(n,k)=(n!/k!)*sum{i=0..n-k, C(n-k-1,i)/(n-k-i)!}
From Peter Bala, May 14 2012 : (Start)
Array is exp(S*(I-S)^(-1)) where S is A132440 the infinitesimal generator for Pascal's triangle.
Column 0 is A000262.
T(n,k) = binomial(n,k)*A000262(n-k).
So T(n,k) gives the number of ways to choose a subset of {1,2,...,n} of size k and then arrange the remaining n-k elements into a set of lists. (End)
T(n,k) = (-1)^(k-n+1)*C(n,k)*KummerU(k-n+1, 2, -1). - Peter Luschny, Sep 17 2014
From Tom Copeland, Mar 11 2016: (Start)
The row polynomials P_n(x) form an Appell sequence with e.g.f. e^(t*P.(x)) = e^[t / (1-t)] e^(x*t), so the lowering and raising operators are L = d/dx = D and the R = x + 1 / (1-D)^2 = x + 1 + 2 D + 3 D^2 + ..., satisfying L P_n(x) = n * P_(n-1)(x) and R P_n(x) = P_(n+1)(x).
(P.(x) + y)^n = Sum_{k=0..n} binomial(n,k) P_k(x) * y^(n-k) = P_n(x+y).
The Appell polynomial umbral compositional inverse sequence has the e.g.f. e^(t*Q.(x)) = e^[-t / (1-t)] e^(x*t) (see A111884 and A133314), so Q_n(P.(x)) = P_n(Q.(x)) = x^n. The lower triangular matrices for the coefficients of these two Appell sequences are a multiplicative inverse pair.
(End)
Sum_{k=0..n} (-1)^k * T(n,k) = A052845(n). - Alois P. Heinz, Feb 21 2022

A293049 Expansion of e.g.f. exp(x^3/(1 - x)).

Original entry on oeis.org

1, 0, 0, 6, 24, 120, 1080, 10080, 100800, 1149120, 14515200, 199584000, 2973801600, 47740492800, 820928908800, 15049152518400, 292919058432000, 6031865968128000, 130990787582054400, 2991455760887193600, 71659101232502784000, 1796424431562528768000

Offset: 0

Seiichi Manyama, Sep 29 2017

nonn

For n > 4, a(n) is a multiple of 10. - Muniru A Asiru, Oct 09 2017

Seiichi Manyama, Table of n, a(n) for n = 0..444

Column k=2 of A293053.
E.g.f.: Product_{i>k} exp(x^i): A000262 (k=0), A052845 (k=1), this sequence (k=2), A293050 (k=3).
Cf. A361533, A361572.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*j!, j=3..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 30 2017
seq(factorial(k)*coeftayl(exp(x^3/(1-x)), x = 0, k),k=0..50); # Muniru A Asiru, Oct 09 2017

CoefficientList[Series[E^(x^3/(1-x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2017 *)

x='x+O('x^66); Vec(serlaplace(exp(x^3/(1-x))))

E.g.f.: Product_{i>2} exp(x^i).
a(n) ~ n^(n-1/4) * exp(-5/2 + 2*sqrt(n) - n) / sqrt(2). - Vaclav Kotesovec, Sep 30 2017
a(n) = 2*(n-1) * a(n-1) - (n-1)*(n-2) * a(n-2) + 6*binomial(n-1,2) * a(n-3) - 12*binomial(n-1,3) * a(n-4) for n > 3. - Seiichi Manyama, Mar 15 2023
From Seiichi Manyama, Jun 17 2024: (Start)
a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-2*k-1,n-3*k)/k!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=3..n} k * a(n-k)/(n-k)!. (End)

A293053 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>k} exp(x^i).

Original entry on oeis.org

1, 1, 1, 1, 0, 3, 1, 0, 2, 13, 1, 0, 0, 6, 73, 1, 0, 0, 6, 36, 501, 1, 0, 0, 0, 24, 240, 4051, 1, 0, 0, 0, 24, 120, 1920, 37633, 1, 0, 0, 0, 0, 120, 1080, 17640, 394353, 1, 0, 0, 0, 0, 120, 720, 10080, 183120, 4596553, 1, 0, 0, 0, 0, 0, 720, 5040, 100800, 2116800, 58941091

Offset: 0

Seiichi Manyama, Sep 29 2017

			Square array begins:
    1,   1,   1,   1, ...
    1,   0,   0,   0, ...
    3,   2,   0,   0, ...
   13,   6,   6,   0, ...
   73,  36,  24,  24, ...
  501, 240, 120, 120, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..3 give A000262, A052845, A293049, A293050.
Rows n=0..1 give A000012, A000007.
Main diagonal gives A000007.
A(n,n-1) gives A000142(n).
Cf. A293024, A293119.

A:= proc(n, k) option remember; `if`(n=0, 1, add(
      A(n-j, k)*binomial(n-1, j-1)*j!, j=1+k..n))
    end:
seq(seq(A(n,d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 29 2017

A[0, ] = 1; A[n, k_] /; n <= k = 0; A[n_, k_] := A[n, k] = Sum[(i+1)! Binomial[n-1, i] A[n-1-i, k], {i, k, n-1}];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Nov 07 2020 *)

def f(n)
  return 1 if n < 2
  (1..n).inject(:*)
end
def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
  ary = [1]
  (1..n).each{|i| ary << (k..i - 1).inject(0){|s, j| s + f(j + 1) * ncr(i - 1, j) * ary[i - 1 - j]}}
  ary
end
def A293053(n)
  a = []
  (0..n).each{|i| a << A(i, n - i)}
  ary = []
  (0..n).each{|i|
    (0..i).each{|j|
      ary << a[i - j][j]
    }
  }
  ary
end
p A293053(20)

E.g.f. of column k: exp(x^(k+1)/(1-x)).
A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = Sum_{i=k..n-1} (i+1)!*binomial(n-1,i)*A(n-1-i,k) for n > k.
A(n,k) = 2*(n-1) * A(n-1,k) - (n-1)*(n-2) * A(n-2,k) + (k+1)!*binomial(n-1,k) * A(n-1-k,k) - k*(k+1)!*binomial(n-1,k+1) * A(n-2-k,k) for n > k+1. - Seiichi Manyama, Mar 15 2023

A088026 Number of "sets of even lists" for even n, cf. A000262.

Original entry on oeis.org

1, 2, 36, 1560, 122640, 15150240, 2695049280, 650948538240, 204637027795200, 81098021561356800, 39516616693678924800, 23204736106751520921600, 16152539421202464036556800, 13145716394493318293898240000, 12363004898960780220305909760000

Offset: 0

Vladeta Jovovic, Nov 02 2003

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A052845, A088009, A206703.

b:= proc(n) option remember; `if`(n=0, 1, add((i->
      b(n-i)*binomial(n-1, i-1)*i!)(2*j), j=1..n/2))
    end:
a:= n-> b(2*n):
seq(a(n), n=0..14);  # Alois P. Heinz, Feb 01 2022

Table[n!*SeriesCoefficient[E^(x^2/(1-x^2)),{x,0,n}],{n,0,40,2}] (* Vaclav Kotesovec, Feb 25 2014 *)

x='x+O('x^66); /* (half) that many terms */
v=Vec(serlaplace(exp(x^2/(1-x^2))));
vector(#v\2,n, v[2*n-1])
/* Joerg Arndt, Jul 29 2012 */

E.g.f.: exp(x^2/(1-x^2)) (even powers only, see PARI code).
E.g.f.: exp(x^2/(1-x^2)) = 4/(2-(x^2/(1-x^2))*G(0))-1 where G(k) = 1 - x^4/(x^4 + 4*(1-x^2)^2*(2*k+1)*(2*k+3)/G(k+1) ) (continued fraction). - Sergei N. Gladkovskii, Dec 10 2012
a(n) ~ 2^(2*n) * n^(2*n-1/4) * exp(sqrt(4*n)-2*n-1/2). - Vaclav Kotesovec, Feb 25 2014
D-finite with recurrence a(n) -2*(2*n-1)^2*a(n-1) +4*(n-1)*(n-2)*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Feb 01 2022
a(n) = A206703(2n,n). - Alois P. Heinz, Feb 19 2022

More terms from Joerg Arndt, Jul 29 2012.

A293050 Expansion of e.g.f. exp(x^4/(1 - x)).

Original entry on oeis.org

1, 0, 0, 0, 24, 120, 720, 5040, 60480, 725760, 9072000, 119750400, 1756339200, 28021593600, 479480601600, 8717829120000, 168254102016000, 3438311804928000, 74160828758016000, 1682757222322176000, 40061786401308672000, 998402161605488640000

Offset: 0

Seiichi Manyama, Sep 29 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..444

Column k=3 of A293053.
E.g.f.: Product_{i>k} exp(x^i): A000262 (k=0), A052845 (k=1), A293049 (k=2), this sequence (k=3).
Cf. A361545, A361576.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*j!, j=4..n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 29 2017

a[n_] := a[n] = If[n==0, 1, Sum[a[n-j] Binomial[n-1, j-1] j!, {j, 4, n}]];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz *)

x='x+O('x^66); Vec(serlaplace(exp(x^4/(1-x))))

E.g.f.: Product_{i>3} exp(x^i).
From Vaclav Kotesovec, Sep 30 2017: (Start)
a(n) = 2*(n-1)*a(n-1) - (n-2)*(n-1)*a(n-2) + 4*(n-3)*(n-2)*(n-1)*a(n-4) - 3*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).
a(n) ~ n^(n-1/4) * exp(-7/2 + 2*sqrt(n) - n) / sqrt(2).
(End)
From Seiichi Manyama, Jun 17 2024: (Start)
a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n-3*k-1,n-4*k)/k!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=4..n} k * a(n-k)/(n-k)!. (End)

A113236 Number of partitions of {1,..,n} into any number of lists of size not equal to 3, where a list means an ordered subset, cf. A000262.

Original entry on oeis.org

1, 1, 3, 7, 49, 321, 2851, 24823, 256257, 2887489, 36759331, 507010791, 7597222513, 122184356737, 2106356007939, 38693238713431, 754792977928321, 15572911248409473, 338800604611562947, 7749991799652960199, 185934065196259734321, 4667877395135551746241

Offset: 0

Karol A. Penson, Oct 19 2005

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..444

Cf. A052845, A113235.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x*(1-x^2+x^3)/(1-x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 17 2018

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=3, 0, a(n-j)*binomial(n-1, j-1)*j!), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, May 10 2016

Range[0, 18]!*CoefficientList[ Series[ Exp[x*(1-x^2+x^3)/(1 - x)], {x, 0, 18}], x] (* Zerinvary Lajos, Mar 23 2007 *)
a[n_] := a[n] = If[n==0, 1, Sum[If[j==3, 0, a[n-j]*Binomial[n-1, j-1]*j!], {j, 1, n}]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 11 2017, after Alois P. Heinz *)

x='x+O('x^30); Vec(serlaplace(exp(x*(1-x^2+x^3)/(1-x)))) \\ G. C. Greubel, May 17 2018

E.g.f.: exp(x*(1-x^2+x^3)/(1-x)).
Expression as a sum involving generalized Laguerre polynomials, in Mathematica notation: a(n)=n!*Sum[(-1)^k*LaguerreL[n - 3*k, -1, -1]/k!, {k, 0, Floor[n/3]}], n=0, 1....
a(n) ~ exp(-3/2+2*sqrt(n)-n)*n^(n-1/4)/sqrt(2). - Vaclav Kotesovec, Jun 22 2013

A293117 E.g.f.: exp(x^2/(x-1)).

Original entry on oeis.org

1, 0, -2, -6, -12, 0, 240, 2520, 21840, 181440, 1481760, 11642400, 80498880, 311351040, -4739454720, -183437654400, -4300276780800, -88851284121600, -1754954007206400, -34107089784768000, -659574028252339200, -12724865943229440000, -244046146272658329600

Offset: 0

Seiichi Manyama, Sep 30 2017

sign

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

Column k=1 of A293119.

Cf. A052845.

a:= proc(n) option remember; `if`(n=0, 1, -add(
      a(n-j)*binomial(n-1, j-1)*j!, j=2..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 30 2017

CoefficientList[Series[E^(-x^2/(1-x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2017 *)

x='x+O('x^66); Vec(serlaplace(exp(x^2/(x-1))))

D-finite with recurrence a(n) = (2*n-2)*a(n-1) - (n^2-n)*a(n-2) + (n^2-3*n+2)*a(n-3). - Robert Israel, Apr 29 2021

A293120 Expansion of e.g.f. exp(x^2/(1+x)).

Original entry on oeis.org

1, 0, 2, -6, 36, -240, 1920, -17640, 183120, -2116800, 26943840, -374220000, 5628934080, -91122071040, 1579034096640, -29155689763200, 571308920582400, -11838533804697600, 258608278645516800, -5938673374272038400, 143003892952893772800

Offset: 0

Seiichi Manyama, Sep 30 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..444

Column k=1 of A293133.
Cf. A111884, A293121.
Cf. A052845.

CoefficientList[Series[E^(x^2/(1+x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2017 *)

x='x+O('x^66); Vec(serlaplace(exp(x^2/(1+x))))

E.g.f.: exp(x^2/(1+x)).
a(n) = (-1)^n * A052845(n).
a(n) ~ (-1)^n * n^(n-1/4) * exp(-3/2 + 2*sqrt(n) - n)/sqrt(2). - Vaclav Kotesovec, Sep 30 2017

cp's OEIS Frontend

A088009 Number of "sets of odd lists", cf. A000262.

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A113235 Number of partitions of {1,..,n} into any number of lists of size not equal to 2, where a list means an ordered subset, cf. A000262.

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A129652 Exponential Riordan array [e^(x/(1-x)),x].

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A293049 Expansion of e.g.f. exp(x^3/(1 - x)).

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A293053 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>k} exp(x^i).

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A088026 Number of "sets of even lists" for even n, cf. A000262.

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A293050 Expansion of e.g.f. exp(x^4/(1 - x)).

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