A000296 -id:A000296 - cp's OEIS frontend

A288268 Expansion of e.g.f.: exp(Sum_{k>=1} (k-1)*x^k/k).

1, 0, 1, 4, 21, 136, 1045, 9276, 93289, 1047376, 12975561, 175721140, 2581284541, 40864292184, 693347907421, 12548540320876, 241253367679185, 4909234733857696, 105394372192969489, 2380337795595885156, 56410454014314490981, 1399496554158060983080

Offset: 0

Seiichi Manyama, Oct 20 2017

Seiichi Manyama, Table of n, a(n) for n = 0..444
Index entries for sequences related to Laguerre polynomials

Cf. A000296, A002720, A008275, A009940, A052852, A052887, A059114, A288269.

l:= func< n, a, b | Evaluate(LaguerrePolynomial(n, a), b) >;
[1,0]cat[(Factorial(n)/(n-1))*(2*l(n-1,0,-1) - l(n,0,-1)): n in [2..30]]; // G. C. Greubel, Mar 10 2021

a := proc(n) option remember; if n < 3 then [1, 0, 1][n+1] else
-(n^2 - 4*n + 3)*a(n - 2) + (2*n - 2)*a(n - 1) fi end:
seq(a(n), n = 0..21); # Peter Luschny, Feb 20 2022

Table[If[n<2, 1-n, (n!/(n-1))*(2*LaguerreL[n-1, -1] - LaguerreL[n, -1])], {n, 0, 30}] (* G. C. Greubel, Mar 10 2021 *)

{a(n) = n!*polcoeff(exp(sum(k=1, n, (k-1)*x^k/k)+x*O(x^n)), n)}

[1-n if n<2 else (factorial(n)/(n-1))*(2*gen_laguerre(n-1,0,-1) - gen_laguerre(n,0,-1)) for n in (0..30)] # G. C. Greubel, Mar 10 2021

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} (k-1)*a(n-k)/(n-k)! for n > 0.
E.g.f.: (1 - x) * exp(x/(1 - x)). - Ilya Gutkovskiy, Jul 27 2020
a(n) = (n!/(n-1))*( 2*LaguerreL(n-1, -1) - LaguerreL(n, -1) ) with a(0) = 1, a(1) = 0. - G. C. Greubel, Mar 10 2021
a(n) ~ n^(n - 3/4) * exp(-1/2 + 2*sqrt(n) - n) / sqrt(2) * (1 - 65/(48*sqrt(n))). - Vaclav Kotesovec, Mar 10 2021, minor term corrected Dec 01 2021
From Peter Luschny, Feb 20 2022: (Start)
a(n) = n! * Sum_{k=0..n} (-1)^k * LaguerreL(n-k, k-1, -1).
a(n) = 2*(n - 1)*a(n - 1) - (n^2 - 4*n + 3)*a(n - 2) for n >= 3. (End)
From Peter Bala, May 26 2023: (Start)
a(n) = Sum_{k = 0..n} |Stirling1(n,k)|*A000296(k) (follows from the fundamental theorem of Riordan arrays).
Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is purely periodic with the period dividing k. For example, modulo 7 we obtain the purely periodic sequence [1, 0, 1, 4, 0, 3, 2, 1, 0, 1, 4, 0, 3, 2, ...] of period 7. Cf. A047974. (End)
For n>1, a(n) = (2*n*A002720(n-1) - A002720(n))/(n-1). - Vaclav Kotesovec, May 27 2023

A343664 Number of partitions of an n-set without blocks of size 4.

Original entry on oeis.org

1, 1, 2, 5, 14, 47, 173, 702, 3125, 14910, 76495, 418035, 2418397, 14791597, 95093612, 641094695, 4521228732, 33250447919, 254585084539, 2024995604762, 16702070759557, 142642458681486, 1259387604241013, 11479967000116911, 107910143688962037, 1044735841257587203, 10407104137208385924

Offset: 0

Ilya Gutkovskiy, Apr 25 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..581

Cf. A000110, A000296, A027338, A097514, A124504, A343665, A343666, A343667, A343668, A343669.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=4, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..26);  # Alois P. Heinz, Apr 25 2021

nmax = 26; CoefficientList[Series[Exp[Exp[x] - 1 - x^4/4!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 4 k]/((n - 4 k)! k! (4!)^k), {k, 0, Floor[n/4]}], {n, 0, 26}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 4, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 26}]

E.g.f.: exp(exp(x) - 1 - x^4/4!).

a(n) = n! * Sum_{k=0..floor(n/4)} (-1)^k * Bell(n-4*k) / ((n-4*k)! * k! * (4!)^k).

A346739 Expansion of e.g.f.: exp(exp(x) - 4*x - 1).

Original entry on oeis.org

1, -3, 10, -35, 127, -472, 1787, -6855, 26572, -103765, 407695, -1608378, 6369117, -25271183, 100542930, -400114103, 1597052419, -6359524256, 25481982047, -101103395443, 409291679676, -1592903606657, 6729506287091, -23748796926026, 123501587468073, -227183793907851

Offset: 0

Ilya Gutkovskiy, Jul 31 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..580

Cf. A000110, A000296, A045379, A126617, A193684, A290219, A346738, A346740.

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(Exp(x) -4*x -1) ))) // G. C. Greubel, Jun 12 2024

nmax = 25; CoefficientList[Series[Exp[Exp[x] - 4 x - 1], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] (-4)^(n - k) BellB[k], {k, 0, n}], {n, 0, 25}]
a[0] = 1; a[n_] := a[n] = -4 a[n - 1] + Sum[Binomial[n - 1, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 25}]

[factorial(n)*( exp(exp(x) -4*x -1) ).series(x, n+1).list()[n] for n in (0..30)] # G. C. Greubel, Jun 12 2024

G.f. A(x) satisfies: A(x) = (1 - x + x * A(x/(1 - x))) / ((1 - x) * (1 + 4*x)).
a(n) = Sum_{k=0..n} binomial(n,k) * (-4)^(n-k) * Bell(k).
a(n) = exp(-1) * Sum_{k>=0} (k - 4)^n / k!.
a(0) = 1; a(n) = -4 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k).

A178866 Basic Multinomial Coefficients.

Original entry on oeis.org

1, 1, 1, 3, 1, 10, 1, 15, 15, 10, 1, 105, 35, 21, 1, 280, 210, 105, 56, 35, 28, 1, 1260, 1260, 378, 280, 126, 84, 36, 1, 6300, 3150, 2520, 2100, 1575, 945, 630, 210, 126, 120, 45, 1, 34650, 17325, 15400, 6930, 6930, 5775, 4620, 4620, 990, 462, 330, 165, 55, 1

Offset: 1

Johannes W. Meijer and Manuel Nepveu (Manuel.Nepveu(AT)tno.nl), Jun 21 2010, Jun 24 2010

All multinomial coefficients (MC's) are equal, but some are more equal than others. These are the basic multinomial coefficients (BMC's). The ordinary multinomial coefficients can be generated with the basic multinomial coefficients; see A178867.
A number n can be partitioned in A000041(n) different ways. The seven partitions of n=5 are e.g. [5] = [1+4] = [2+3] = [1+1+3] = [1+2+2] = [1+1+1+2] = [1+1+1+1+1]. We observe that the k-th partition of n will consist of a certain number of 1s (i.e., "singles"), a certain number of 2s (i.e., "pairs"), a certain number of 3s (i.e., "triples"), a certain number of 4s (i.e., "4-tuples") and so on. We denote with qt the number of t-tuples in the k-th partition of n. We observe that for the third partition of n=5 there is one pair (q2=1) and one triple (q3=1).
The multinomial coefficients are defined by M3[n,k] = n!/product((t!)^qt*(qt)!, t = 1..n), see Abramowitz and Stegun. For the third partition M3[5,3] = 10, so there are 10 different ways of distributing one pair (B1, B1) and one triple (B2, B2, B2) over five positions.
We define the BMC's as the multinomial coefficients M3[n,k] for which there are no singles (q1=0) in the k-th partition of n for n>=2. Furthermore we define a(1) = 1.
The number of a(n) terms in a triangle row lead to sequence A002865(n) (n>=2). The row sums lead to sequence A000296(n) (n>=2).

			The first few triangle rows are (P = Pair; T = Triple; 4-T = 4 Tuple; etc..):
n = 1: 1;
n = 2: 1 (1*P);
n = 3: 1 (1*T);
n = 4: 3 (2*P), 1 (1*4-T);
n = 5: 10 (1*P+1*T), 1 (1*5-T);
n = 6: 15 (3*P), 15 (1*4-T+1*P), 10 (2*T), 1 (1*6-T);
n = 7: 105 (1*T+2*P), 35 (1*4-T+1*T), 21 (1*5-T+1*P), 1 (1*7-T);

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 24, pp. 822-832.
Kevin Brown, The Collector's Problem
J. W. Meijer and M. Nepveu, Euler's ship on the Pentagonal Sea, Acta Nova, Volume 4, No.1, December 2008. pp. 176-187.

Cf. A036040 (version 1), A080575 (version 2) and A178867 (version 3).

with(combinat): nmax:=11; A178866(1):=1: T:=1: for n from 1 to nmax do y(n):=numbpart(n): P(n):=sort(partition(n)): k:=0: for r from 1 to y(n) do if P(n)[r,1]>1 then k:=k+1; B(k):=P(n)[r]: fi; od: A002865(n):=k; for k from 1 to A002865(n) do s:=0: j:=1: while sA002865(n))], `>`): for k from 1 to A002865(n) do M3[n,k]:=a[k] od: for k from 1 to A002865(n) do T:=T+1: A178866(T):= M3[n,k]: od: od: seq(A178866(n),n=1..T);

n = sum(qt*t, t=1..n)

M3[n,k] = n!/product((t!)^qt*(qt)!, t = 1..n)

A186759 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k cycles that are either nonincreasing or of length 1 (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) < b(2) < b(3) < ... .

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 4, 0, 1, 4, 9, 10, 0, 1, 11, 53, 35, 20, 0, 1, 41, 280, 268, 95, 35, 0, 1, 162, 1804, 1904, 903, 210, 56, 0, 1, 715, 12971, 15727, 8008, 2408, 406, 84, 0, 1, 3425, 104600, 142533, 80323, 25662, 5502, 714, 120, 0, 1, 17722, 936370, 1418444, 871575, 303385, 68712, 11256, 1170, 165, 0, 1

Offset: 0

Emeric Deutsch, Feb 26 2011

			T(3,1) = 4 because we have (1)(23), (12)(3), (13)(2), and (132).
T(4,4) = 1 because we have (1)(2)(3)(4).
Triangle starts:
   1;
   0,  1;
   1,  0,  1;
   1,  4,  0,  1;
   4,  9, 10,  0, 1;
  11, 53, 35, 20, 0, 1;

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A000296, A186754, A186755, A186756, A186757, A186758, A186760.

G := exp((1-t)*(exp(z)-1-z))/(1-z)^t: 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, j), j = 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-i)*binomial(n-1, i-1)*
      `if`(i=1, x, 1+x*((i-1)!-1)), i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Sep 25 2016

b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n-i]*Binomial[n-1, i-1]*If[i == 1, x, 1+x*((i-1)!-1)], {i, 1, n}]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Nov 28 2016 after Alois P. Heinz *)

E.g.f.: G(t,z) = exp((1-t)*(exp(z)-1-z))/(1-z)^t.
The 4-variate e.g.f. H(u,v,w,z) of the permutations of {1,2,...,n} with respect to size (marked by z), number of fixed points (marked by u), number of increasing cycles of length >=2 (marked by v), and number of nonincreasing cycles (marked by w) is given by H(u,v,w,z) = exp(u*z+v*(exp(z)-1-z)+w*(1-exp(z)))/(1-z)^w. Remark: the nonincreasing cycles are necessarily of length >=3. We have: G(t,z) = H(t,1,t,z).
Sum_{k=0..n} T(n,k) = n!.
T(n,0) = A000296(n).
Sum_{k=0..n} k*T(n,k) = A186760(n).

A306417 Number of self-conjugate set partitions of {1, ..., n}.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 7, 7, 46, 39, 321

Offset: 0

Gus Wiseman, Feb 14 2019

This sequence counts set partitions fixed under Callan's conjugation operation.

			The  a(3) = 1 through a(7) = 7 self-conjugate set partitions:
  {{12}{3}}  {{13}{24}}  {{123}{4}{5}}  {{135}{246}}    {{13}{246}{57}}
                         {{13}{2}{45}}  {{124}{35}{6}}  {{15}{246}{37}}
                                        {{13}{25}{46}}  {{1234}{5}{6}{7}}
                                        {{14}{2}{356}}  {{124}{3}{56}{7}}
                                        {{14}{236}{5}}  {{134}{2}{5}{67}}
                                        {{14}{25}{36}}  {{14}{2}{3}{567}}
                                        {{145}{26}{3}}  {{14}{23}{57}{6}}

David Callan, On conjugates for set partitions and integer compositions, arXiv:math/0508052 [math.CO], 2005.

Cf. A000110, A000126, A000296, A001610, A032032, A052841, A080107, A169985, A306416, A324011, A324012.

A324012 Number of self-complementary set partitions of {1, ..., n} with no singletons or cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 3, 2, 14, 11, 80, 85, 510

Offset: 0

Gus Wiseman, Feb 12 2019

The complement of a set partition pi of {1, ..., n} is defined as n + 1 - pi (elementwise) on page 3 of Callan. For example, the complement of {{1,5},{2},{3,6},{4}} is {{1,4},{2,6},{3},{5}}. This sequence counts certain self-conjugate set partitions, i.e., fixed points under Callan's conjugation operation.

			The  a(6) = 3 through a(9) = 11 self-complementary set partitions with no singletons or cyclical adjacencies:
  {{135}{246}}    {{13}{246}{57}}  {{1357}{2468}}      {{136}{258}{479}}
  {{13}{25}{46}}  {{15}{246}{37}}  {{135}{27}{468}}    {{147}{258}{369}}
  {{14}{25}{36}}                   {{146}{27}{358}}    {{148}{269}{357}}
                                   {{147}{258}{36}}    {{168}{249}{357}}
                                   {{157}{248}{36}}    {{13}{258}{46}{79}}
                                   {{13}{24}{57}{68}}  {{14}{258}{37}{69}}
                                   {{13}{25}{47}{68}}  {{14}{28}{357}{69}}
                                   {{14}{26}{37}{58}}  {{16}{258}{37}{49}}
                                   {{14}{27}{36}{58}}  {{16}{28}{357}{49}}
                                   {{15}{26}{37}{48}}  {{17}{258}{39}{46}}
                                   {{15}{27}{36}{48}}  {{18}{29}{357}{46}}
                                   {{16}{24}{38}{57}}
                                   {{16}{25}{38}{47}}
                                   {{17}{28}{35}{46}}

David Callan, On conjugates for set partitions and integer compositions, arXiv:math/0508052 [math.CO], 2005.

Cf. A000110, A000126, A000296, A001610, A080107, A169985, A261139, A306417 (all self-conjugate set partitions), A324011 (self-complementarity not required), A324013 (adjacencies allowed), A324014 (singletons allowed), A324015.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
cmp[stn_]:=Union[Sort[Max@@Join@@stn+1-#]&/@stn];
Table[Select[sps[Range[n]],And[cmp[#]==Sort[#],Count[#,{_}]==0,Total[If[First[#]==1&&Last[#]==n,1,0]+Count[Subtract@@@Partition[#,2,1],-1]&/@#]==0]&]//Length,{n,0,10}]

A337038 a(n) = exp(-1/2) * Sum_{k>=0} (2k - 1)^n / (2^k k!).

Original entry on oeis.org

1, 0, 2, 4, 20, 96, 552, 3536, 25104, 194816, 1637408, 14792768, 142761280, 1464117760, 15886137984, 181667507456, 2182268117248, 27456279388160, 360872502280704, 4943580063237120, 70437638474568704, 1041911242274562048, 15972832382065977344, 253388070573020401664

Offset: 0

Ilya Gutkovskiy, Aug 12 2020

nonn

Robert Israel, Table of n, a(n) for n = 0..516

Cf. A000296, A004211, A007405, A124311, A166922, A217203, A337039, A337040, A337041, A337042, A337043.

E:= exp((exp(2*x)-1)/2-x):
S:= series(E,x,31):
seq(coeff(S,x,i)*i!,i=0..30); # Robert Israel, Aug 26 2020

nmax = 23; CoefficientList[Series[Exp[(Exp[2 x] - 1)/2 - x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k] 2^k a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 2^k BellB[k, 1/2], {k, 0, n}], {n, 0, 23}]

G.f. A(x) satisfies: A(x) = (1 - 2*x + x*A(x/(1 - 2*x))) / (1 - x - 2*x^2).
G.f.: (1/(1 + x)) * Sum_{k>=0} (x/(1 + x))^k / Product_{j=1..k} (1 - 2*j*x/(1 + x)).
E.g.f.: exp((exp(2*x) - 1) / 2 - x).
a(0) = 1; a(n) = Sum_{k=1..n-1} binomial(n-1,k) * 2^k * a(n-k-1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A004211(k).
a(n) ~ 2^(n - 1/2) * n^(n - 1/2) * exp(n/LambertW(2*n) - n - 1/2) / (sqrt(1 + LambertW(2*n)) * LambertW(2*n)^(n - 1/2)). - Vaclav Kotesovec, Jun 26 2022

A343665 Number of partitions of an n-set without blocks of size 5.

Original entry on oeis.org

1, 1, 2, 5, 15, 51, 197, 835, 3860, 19257, 102997, 586170, 3535645, 22496437, 150454918, 1054235150, 7718958995, 58905868192, 467530598983, 3851775136517, 32881385742460, 290387471713872, 2649226725182823, 24934118754400767, 241809265181914545, 2413608066257526577

Offset: 0

Ilya Gutkovskiy, Apr 25 2021

nonn

Cf. A000110, A000296, A027339, A097514, A124504, A343664, A343666, A343667, A343668, A343669.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=5, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 25 2021

nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^5/5!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 5 k]/((n - 5 k)! k! (5!)^k), {k, 0, Floor[n/5]}], {n, 0, 25}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 5, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

E.g.f.: exp(exp(x) - 1 - x^5/5!).

a(n) = n! * Sum_{k=0..floor(n/5)} (-1)^k * Bell(n-5*k) / ((n-5*k)! * k! * (5!)^k).

A343666 Number of partitions of an n-set without blocks of size 6.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 202, 870, 4084, 20727, 112825, 654546, 4026487, 26145511, 178550986, 1278168860, 9564026947, 74615547996, 605593775899, 5103054929621, 44564754448972, 402677613100491, 3759094788129312, 36205919126040190, 359340174509911325, 3670825700549853053

Offset: 0

Ilya Gutkovskiy, Apr 25 2021

nonn

Cf. A000110, A000296, A027340, A097514, A124504, A343664, A343665, A343667, A343668, A343669.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=6, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 25 2021

nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^6/6!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 6 k]/((n - 6 k)! k! (6!)^k), {k, 0, Floor[n/6]}], {n, 0, 25}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 6, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

E.g.f.: exp(exp(x) - 1 - x^6/6!).

a(n) = n! * Sum_{k=0..floor(n/6)} (-1)^k * Bell(n-6*k) / ((n-6*k)! * k! * (6!)^k).

cp's OEIS Frontend

A288268 Expansion of e.g.f.: exp(Sum_{k>=1} (k-1)*x^k/k).

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A343664 Number of partitions of an n-set without blocks of size 4.

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A346739 Expansion of e.g.f.: exp(exp(x) - 4*x - 1).

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A178866 Basic Multinomial Coefficients.

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A306417 Number of self-conjugate set partitions of {1, ..., n}.

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A324012 Number of self-complementary set partitions of {1, ..., n} with no singletons or cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

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A337038 a(n) = exp(-1/2) * Sum_{k>=0} (2*k - 1)^n / (2^k * k!).

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A337038 a(n) = exp(-1/2) * Sum_{k>=0} (2k - 1)^n / (2^k k!).