A052841 -id:A052841 - cp's OEIS frontend

A336102 Number of inseparable multisets of size n covering an initial interval of positive integers.

0, 0, 1, 1, 3, 3, 8, 8, 20, 20, 48, 48, 112, 112, 256, 256, 576, 576, 1280, 1280, 2816, 2816, 6144, 6144, 13312, 13312, 28672, 28672, 61440, 61440, 131072, 131072, 278528, 278528, 589824, 589824, 1245184, 1245184, 2621440, 2621440, 5505024, 5505024, 11534336

Offset: 0

Gus Wiseman, Jul 08 2020

nonn

A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.
Alternatively, a multiset is separable if its greatest multiplicity is greater than the sum of its remaining multiplicities plus one.
Also the number of compositions of n whose greatest part is greater than the sum of its remaining parts plus one. For example, the a(2) = 1 through a(7) = 8 compositions are:
  (2)  (3)  (4)    (5)    (6)      (7)
            (1,3)  (1,4)  (1,5)    (1,6)
            (3,1)  (4,1)  (2,4)    (2,5)
                          (4,2)    (5,2)
                          (5,1)    (6,1)
                          (1,1,4)  (1,1,5)
                          (1,4,1)  (1,5,1)
                          (4,1,1)  (5,1,1)

			The a(2) = 1 through a(7) = 8 multisets:
  {11}  {111}  {1111}  {11111}  {111111}  {1111111}
               {1112}  {11112}  {111112}  {1111112}
               {1222}  {12222}  {111122}  {1111122}
                                {111123}  {1111123}
                                {112222}  {1122222}
                                {122222}  {1222222}
                                {122223}  {1222223}
                                {123333}  {1233333}

Michael De Vlieger, Table of n, a(n) for n = 0..6625
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-4).

The strong (weakly decreasing multiplicities) case is A025065.
The bisection is A049610.
The separable version is A336103.
Sequences covering an initial interval are A000670.
Anti-run compositions are A003242.
Anti-run patterns are A005649.
Separable partitions are A325534.
Inseparable partitions are A325535.
Inseparable factorizations are A333487.
Anti-run compositions are ranked by A333489.
Heinz numbers of inseparable partitions are A335448.
Cf. A001792, A019472, A052841, A106351, A124767, A269134, A292884, A335433, A335126, A335452, A335548.

Table[Length[Join@@Permutations/@Select[IntegerPartitions[n],With[{mx=Max@@#},mx>1+Total[DeleteCases[#,mx,{1},1]]]&]],{n,0,15}]
(* Second program: *)
CoefficientList[Series[x^2*(1 - x) (x + 1)^2/(2 x^2 - 1)^2, {x, 0, 43}], x] (* Michael De Vlieger, Apr 07 2021 *)

a(2*n) = a(2*n + 1) = A049610(n + 1).
a(n) = 2^(n-1) - A336103(n).
A001792 repeated for n > 1. David A. Corneth, Jul 09 2020
From Chai Wah Wu, Apr 07 2021: (Start)
a(n) = 4*a(n-2) - 4*a(n-4) for n > 5.
G.f.: x^2*(1 - x)*(x + 1)^2/(2*x^2 - 1)^2. (End)

A349050 Number of multisets of size n that have no alternating permutations and cover an initial interval of positive integers.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 8, 12, 20, 32, 48, 80, 112, 192, 256, 448, 576, 1024, 1280, 2304, 2816, 5120, 6144, 11264, 13312, 24576, 28672, 53248, 61440, 114688, 131072, 245760, 278528, 524288, 589824, 1114112, 1245184, 2359296, 2621440, 4980736, 5505024

Offset: 0

Gus Wiseman, Dec 12 2021

nonn

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). Alternating permutations of multisets are a generalization of alternating or up-down permutations of {1..n}.

			The multiset {1,2,2,2,2,3,3} has no alternating permutations, even though it does have the three anti-run permutations (2,1,2,3,2,3,2), (2,3,2,1,2,3,2), (2,3,2,3,2,1,2), so is counted under a(7).
The a(2) = 1 through a(7) = 12 multisets:
  {11}  {111}  {1111}  {11111}  {111111}  {1111111}
               {1112}  {11112}  {111112}  {1111112}
               {1222}  {12222}  {111122}  {1111122}
                       {12223}  {111123}  {1111123}
                                {112222}  {1122222}
                                {122222}  {1122223}
                                {122223}  {1222222}
                                {123333}  {1222223}
                                          {1222233}
                                          {1222234}
                                          {1233333}
                                          {1233334}
As compositions:
  (2)  (3)  (4)    (5)      (6)      (7)
            (1,3)  (1,4)    (1,5)    (1,6)
            (3,1)  (4,1)    (2,4)    (2,5)
                   (1,3,1)  (4,2)    (5,2)
                            (5,1)    (6,1)
                            (1,1,4)  (1,1,5)
                            (1,4,1)  (1,4,2)
                            (4,1,1)  (1,5,1)
                                     (2,4,1)
                                     (5,1,1)
                                     (1,1,4,1)
                                     (1,4,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Frankie Mennicucci, On the Number of Strong Dominating Sets in a Graph, Master's Thesis, Montclair State Univ. (2024). See p. 32.
Wikipedia, Alternating permutation

The case of weakly decreasing multiplicities is A025065.
The inseparable case is A336102.
A separable instead of alternating version is A336103.
The version for partitions is A345165.
The version for factorizations is A348380, complement A348379.
The complement (still covering an initial interval) is counted by A349055.
A000670 counts sequences covering an initial interval, anti-run A005649.
A001250 counts alternating permutations, complement A348615.
A003242 counts Carlitz (anti-run) compositions, ranked by A333489.
A025047 = alternating compositions, ranked by A345167, also A025048/A025049.
A049774 counts permutations avoiding the consecutive pattern (1,2,3).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A344654 counts partitions w/o an alternating permutation, ranked by A344653.
Cf. A019472, A052841, A096441, A106351, A106356, A335125, A335126, A344614, A347050, A347706, A348377, A349054.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[allnorm[n],Select[Permutations[#],wigQ]=={}&]],{n,0,7}]

a(n) = if(n==0, 0, if(n%2==0, (n+2)*2^(n/2-3), (n-1)*2^((n-1)/2-2))) \\ Andrew Howroyd, Jan 13 2024

a(n) = A011782(n) - A349055(n).

a(n) = (n+2)*2^(n/2-3) for even n > 0; a(n) = (n-1)*2^((n-5)/2) for odd n. - Andrew Howroyd, Jan 13 2024

Terms a(10) and beyond from Andrew Howroyd, Jan 13 2024

A340602 Heinz numbers of integer partitions of even rank.

Original entry on oeis.org

1, 2, 5, 6, 8, 9, 11, 14, 17, 20, 21, 23, 24, 26, 30, 31, 32, 35, 36, 38, 39, 41, 44, 45, 47, 49, 50, 54, 56, 57, 58, 59, 65, 66, 67, 68, 73, 74, 75, 80, 81, 83, 84, 86, 87, 91, 92, 95, 96, 97, 99, 102, 103, 104, 106, 109, 110, 111, 120, 122, 124, 125, 126, 127

Offset: 1

Gus Wiseman, Jan 21 2021

nonn

The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is 0.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The sequence of partitions with their Heinz numbers begins:
     1: ()           31: (11)           58: (10,1)
     2: (1)          32: (1,1,1,1,1)    59: (17)
     5: (3)          35: (4,3)          65: (6,3)
     6: (2,1)        36: (2,2,1,1)      66: (5,2,1)
     8: (1,1,1)      38: (8,1)          67: (19)
     9: (2,2)        39: (6,2)          68: (7,1,1)
    11: (5)          41: (13)           73: (21)
    14: (4,1)        44: (5,1,1)        74: (12,1)
    17: (7)          45: (3,2,2)        75: (3,3,2)
    20: (3,1,1)      47: (15)           80: (3,1,1,1,1)
    21: (4,2)        49: (4,4)          81: (2,2,2,2)
    23: (9)          50: (3,3,1)        83: (23)
    24: (2,1,1,1)    54: (2,2,2,1)      84: (4,2,1,1)
    26: (6,1)        56: (4,1,1,1)      86: (14,1)
    30: (3,2,1)      57: (8,2)          87: (10,2)

FindStat, St000145: The Dyson rank of a partition

Taking only length gives A001222.
Taking only maximum part gives A061395.
These partitions are counted by A340601.
The complement is A340603.
The case of positive rank is A340605.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A101198 counts partitions of rank 1 (A325233).
A101707 counts partitions of odd positive rank (A340604).
A101708 counts partitions of even positive rank (A340605).
A257541 gives the rank of the partition with Heinz number n.
A324516 counts partitions with rank = maximum minus minimum part (A324515).
A340653 counts factorizations of rank 0.
A340692 counts partitions of odd rank (A340603).
- Even -
A024430 counts set partitions of even length.
A027187 counts partitions of even length (A028260).
A027187 (also) counts partitions of even maximum (A244990).
A034008 counts compositions of even length.
A035363 counts partitions into even parts (A066207).
A052841 counts ordered set partitions of even length.
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts even-length partitions of even numbers (A340784).
A339846 counts factorizations of even length.
Cf. A000041, A006141, A056239, A072233, A112798, A168659, A325134, A326836, A326845, A340386, A340387.

Select[Range[100],EvenQ[PrimePi[FactorInteger[#][[-1,1]]]-PrimeOmega[#]]&]

Either n = 1 or A061395(n) - A001222(n) is even.

A349055 Number of multisets of size n that have an alternating permutation and cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 1, 3, 5, 12, 24, 52, 108, 224, 464, 944, 1936, 3904, 7936, 15936, 32192, 64512, 129792, 259840, 521472, 1043456, 2091008, 4183040, 8375296, 16752640, 33525760, 67055616, 134156288, 268320768, 536739840, 1073496064, 2147205120, 4294443008, 8589344768

Offset: 0

Gus Wiseman, Dec 12 2021

nonn

The multisets that have an alternating permutation are those which have no part with multiplicity greater than floor(n/2) except for odd n when either the smallest or largest part can have multiplicity ceiling(n/2). - Andrew Howroyd, Jan 13 2024

			The multiset {1,2,2,3} has alternating permutations (2,1,3,2), (2,3,1,2), so is counted under a(4).
The a(1) = 1 through a(5) = 12 multisets:
  {1}  {1,2}  {1,1,2}  {1,1,2,2}  {1,1,1,2,2}
              {1,2,2}  {1,1,2,3}  {1,1,1,2,3}
              {1,2,3}  {1,2,2,3}  {1,1,2,2,2}
                       {1,2,3,3}  {1,1,2,2,3}
                       {1,2,3,4}  {1,1,2,3,3}
                                  {1,1,2,3,4}
                                  {1,2,2,3,3}
                                  {1,2,2,3,4}
                                  {1,2,3,3,3}
                                  {1,2,3,3,4}
                                  {1,2,3,4,4}
                                  {1,2,3,4,5}
As compositions:
  (1)  (1,1)  (1,2)    (2,2)      (2,3)
              (2,1)    (1,1,2)    (3,2)
              (1,1,1)  (1,2,1)    (1,1,3)
                       (2,1,1)    (1,2,2)
                       (1,1,1,1)  (2,1,2)
                                  (2,2,1)
                                  (3,1,1)
                                  (1,1,1,2)
                                  (1,1,2,1)
                                  (1,2,1,1)
                                  (2,1,1,1)
                                  (1,1,1,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Wikipedia, Alternating permutation

The strong inseparable case is A025065.
A separable instead of alternating version is A336103, complement A336102.
The case of weakly decreasing multiplicities is A336106.
The version for non-twin partitions is A344654, ranked by A344653.
The complement for non-twin partitions is A344740, ranked by A344742.
The complement for partitions is A345165, ranked by A345171.
The version for partitions is A345170, ranked by A345172.
The version for factorizations is A348379, complement A348380.
The complement (still covering an initial interval) is counted by A349050.
A000670 counts sequences covering an initial interval, anti-run A005649.
A001250 counts alternating permutations, complement A348615.
A003242 counts Carlitz (anti-run) compositions, ranked by A333489.
A025047 = alternating compositions, ranked by A345167, also A025048/A025049.
A049774 counts permutations avoiding the consecutive pattern (1,2,3).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
Cf. A019472, A052841, A096441, A106351, A106356, A335125, A335127, A344614, A347050, A347706, A348610, A348613, A349054.

allnorm[n_]:=If[n<=0,{{}},Function[s, Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
wigQ[y_]:=Or[Length[y]==0, Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[allnorm[n], Select[Permutations[#],wigQ]!={}&]],{n,0,7}]

a(n) = if(n==0, 1, 2^(n-1) - if(n%2==0, (n+2)*2^(n/2-3), (n-1)*2^((n-5)/2))) \\ Andrew Howroyd, Jan 13 2024

a(n) = A011782(n) - A349050(n).

a(n) = 2^(n-1) - (n+2)*2^(n/2-3) for even n > 0; a(n) = 2^(n-1) - (n-1)*2^((n-5)/2) for odd n. - Andrew Howroyd, Jan 13 2024

Terms a(10) and beyond from Andrew Howroyd, Jan 13 2024

A154921 Triangle read by rows, T(n, k) = binomial(n, k) * Sum_{j=0..n-k} E(n-k, j)*2^j, where E(n, k) are the Eulerian numbers A173018(n, k), for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 13, 9, 3, 1, 75, 52, 18, 4, 1, 541, 375, 130, 30, 5, 1, 4683, 3246, 1125, 260, 45, 6, 1, 47293, 32781, 11361, 2625, 455, 63, 7, 1, 545835, 378344, 131124, 30296, 5250, 728, 84, 8, 1, 7087261, 4912515, 1702548, 393372, 68166, 9450, 1092, 108, 9, 1

Offset: 0

Mats Granvik, Jan 17 2009

Previous name: Matrix inverse of A154926.
A000670 appears in the first column. A052882 appears in the second column. A000027 and A045943 appear as diagonals. An alternative to calculating the matrix inverse of A154926 is to move the term in the lower right corner to a position in the same column and calculate the determinant instead, which yields the same answer.
Matrix inverse of (2*I - P), where P is Pascal's triangle and I the identity matrix. See A162312 for the matrix inverse of (2*P - I) and some general remarks about arrays of the form M(a) := (I - a*P)^-1 and their connection with weighted sums of powers of integers. The present array equals (1/2)*M(1/2). - Peter Bala, Jul 01 2009
From Mats Granvik, Aug 11 2009: (Start)
The values in this triangle can be seen as permanents of the Pascal triangle analogous to the method in the Redheffer matrix. The elements satisfy (T(n,k)/T(n,k-1))*k = (T(n-1,k)/T(n,k))*n which converges to log(2) as n->oo and k->0. More generally to calculate log(x) multiply the negative values in A154926 by 1/(x-1) and calculate the matrix inverse. Then (T(n,k)/T(n,k-1))*k and (T(n-1,k)/T(n,k))*n in the resulting triangle converge to log(x).
This method for calculating log(x) converges faster than the Taylor series when x is greater than 5 or so. See chapter on Taylor series in Spiegel for comparison. (End)
Exponential Riordan array [1/(2-exp(x)),x]. - Paul Barry, Apr 06 2011
T(n,k) is the number of ordered set partitions of {1,2,...,n} such that the first block contains k elements.  For k=0 the first block contains arbitrarily many elements. - Geoffrey Critzer, Jul 22 2013
A natural (signed) refinement of these polynomials is given by the Appell sequence e.g.f. e^(xt)/ f(t) = exp[tP.(x)] with the formal Taylor series f(x) = 1 + x[1] x + x[2] x^2/2! + ... and with raising operator R = x - d[log(f(D)]/dD (cf. A263634). - Tom Copeland, Nov 06 2015

			From _Peter Bala_, Jul 01 2009: (Start)
Triangle T(n, k) begins:
n\k|     0     1     2     3     4     5     6
==============================================
0  |     1
1  |     1     1
2  |     3     2     1
3  |    13     9     3     1
4  |    75    52    18     4     1
5  |   541   375   130    30     5     1
6  |  4683  3246  1125   260    45     6     1
...
(End)
From _Mats Granvik_, Aug 11 2009: (Start)
Row 4 equals 75,52,18,4,1 because permanents of:
  1,0,0,0,1  1,0,0,0,0  1,0,0,0,0  1,0,0,0,0  1,0,0,0,0
  1,1,0,0,0  1,1,0,0,1  1,1,0,0,0  1,1,0,0,0  1,1,0,0,0
  1,2,1,0,0  1,2,1,0,0  1,2,1,0,1  1,2,1,0,0  1,2,1,0,0
  1,3,3,1,0  1,3,3,1,0  1,3,3,1,0  1,3,3,1,1  1,3,3,1,0
  1,4,6,4,0  1,4,6,4,0  1,4,6,4,0  1,4,6,4,0  1,4,6,4,1
are:
     75         52         18          4          1
(End)

Murray R. Spiegel, Mathematical handbook, Schaum's Outlines, p. 111.

Alois P. Heinz, Rows n = 0..140, flattened
R. B. Nelsen, Problem E3062, Amer. Math. Monthly, Vol. 94, No. 4 (Apr., 1987), 376-377.
R. B. Nelsen and H. Schmidt, Jr., Chains in Power Sets, Mathematics Magazine, Vol. 64, No. 1 (Feb., 1991), 23-31.

Cf. A000629 (row sums), A000670, A007047, A052822 (column 1), A052841 (alt. row sums), A080253, A162312, A162313.

Cf. A263634, A099880 (T(2n,n)).

A154921_row := proc(n) local i,p; p := proc(n,x) option remember; local k;
if n = 0 then 1 else add(p(k,0)*binomial(n,k)*(1+x^(n-k)),k=0..n-1) fi end:
seq(coeff(p(n,x),x,i),i=0..n) end: for n from 0 to 5 do A154921_row(n) od;
# Peter Luschny, Jul 15 2012
T := (n,k) -> binomial(n,k)*add(combinat:-eulerian1(n-k,j)*2^j, j=0..n-k):
seq(print(seq(T(n,k), k=0..n)),n=0..6); # Peter Luschny, Feb 07 2015
# third Maple program:
b:= proc(n) b(n):= `if`(n=0, 1, add(b(n-j)/j!, j=1..n)) end:
T:= (n, k)-> n!/k! *b(n-k):
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Feb 03 2019
# fourth Maple program:
p := proc(n, m) option remember; if n = 0 then 1 else
    (m + x)*p(n - 1, m) + (m + 1)*p(n - 1, m + 1) fi end:
row := n -> local k; seq(coeff(p(n, 0), x, k), k = 0..n):
for n from 0 to 6 do row(n) od;  # Peter Luschny, Jun 23 2023

nn = 8; a = Exp[x] - 1;
Map[Select[#, # > 0 &] &,
  Transpose[
   Table[Range[0, nn]! CoefficientList[
Series[x^n/n!/(1 - a), {x, 0, nn}], x], {n, 0, nn}]]] // Grid (* Geoffrey Critzer, Jul 22 2013 *)
E1[n_ /; n >= 0, 0] = 1; E1[n_, k_] /; k < 0 || k > n = 0; E1[n_, k_] := E1[n, k] = (n - k) E1[n - 1, k - 1] + (k + 1) E1[n - 1, k];
T[n_, k_] := Binomial[n, k] Sum[E1[n - k, j] 2^j, {j, 0, n - k}];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 30 2018, after Peter Luschny *)

@CachedFunction
def Poly(n, x):
    return 1 if n == 0 else add(Poly(k,0)*binomial(n,k)*(x^(n-k)+1) for k in range(n))
R = PolynomialRing(ZZ, 'x')
for n in (0..6): print(R(Poly(n,x)).list()) # Peter Luschny, Jul 15 2012

From Peter Bala, Jul 01 2009: (Start)
TABLE ENTRIES
  (1) T(n,k) = binomial(n,k)*A000670(n-k).
GENERATING FUNCTION
  (2) exp(x*t)/(2-exp(t)) = 1 + (1+x)*t + (3+2*x+x^2)*t^2/2! + ....
PROPERTIES OF THE ROW POLYNOMIALS
The row generating polynomials R_n(x) form an Appell sequence. They appear in the study of the poset of power sets [Nelsen and Schmidt].
The first few values are R_0(x) = 1, R_1(x) = 1+x, R_2(x) = 3+2*x+x^2 and R_3(x) = 13+9*x+3*x^2+x^3.
The row polynomials may be recursively computed by means of
  (3) R_n(x) = x^n + Sum_{k = 0..n-1} binomial(n,k)*R_k(x).
Explicit formulas include
  (4) R_n(x) = (1/2)*Sum_{k >= 0} (1/2)^k*(x+k)^n,
  (5) R_n(x) = Sum_{j = 0..n} Sum_{k = 0..j} (-1)^(j-k)*binomial(j,k) *(x+k)^n,
and
  (6) R_n(x) = Sum_{j = 0..n} Sum_{k = j..n} k!*Stirling2(n,k) *binomial(x,k-j).
SUMS OF POWERS OF INTEGERS
The row polynomials satisfy the difference equation
  (7) 2*R_m(x) - R_m(x+1) = x^m,
which easily leads to the evaluation of the weighted sums of powers of integers
  (8) Sum_{k = 1..n-1} (1/2)^k*k^m = 2*R_m(0) - (1/2)^(n-1)*R_m(n).
For example, m = 2 gives
  (9) Sum_{k = 1..n-1} (1/2)^k*k^2 = 6 - (1/2)^(n-1)*(n^2+2*n+3).
More generally we have
  (10) Sum_{k=0..n-1} (1/2)^k*(x+k)^m = 2*R_m(x) - (1/2)^(n-1)*R_m(x+n).
RELATIONS WITH OTHER SEQUENCES
Sequences in the database given by particular values of the row polynomials are
  (11) A000670(n) = R_n(0)
  (12) A052841(n) = R_n(-1)
  (13) A000629(n) = R_n(1)
  (14) A007047(n) = R_n(2)
  (15) A080253(n) = 2^n*R_n(1/2).
This last result is the particular case (x = 0) of the result that the polynomials 2^n*R_n(1/2+x/2) are the row generating polynomials for A162313.
The above formulas should be compared with those for A162312. (End)
From Peter Luschny, Jul 15 2012: (Start)
  (16) A151919(n) = R_n(1/3)*3^n*(-1)^n
  (17) A052882(n) = [x^1] R_n(x)
  (18) A045943(n) = [x^(n-1)] R_n+1(x)
  (19) A099880(n) = [x^n] R_2n(x). (End)
The coefficients in ascending order of x^i of the polynomials p{0}(x) = 1 and p{n}(x) = Sum_{k=0..n-1} binomial(n,k)*p{k}(0)*(1+x^(n-k)). - Peter Luschny, Jul 15 2012

New name by Peter Luschny, Feb 07 2015

A052558 a(n) = n! ((-1)^n + 2n + 3)/4.

Original entry on oeis.org

1, 1, 4, 12, 72, 360, 2880, 20160, 201600, 1814400, 21772800, 239500800, 3353011200, 43589145600, 697426329600, 10461394944000, 188305108992000, 3201186852864000, 64023737057280000, 1216451004088320000, 26761922089943040000, 562000363888803840000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Stirling transform of (-1)^(n+1)*a(n-1) = [1, -1, 4, -12, 72, -360, ...] is A052841(n-1) = [1,0,2,6,38,270,...]. - Michael Somos, Mar 04 2004
The Stirling transform of this sequence is A258369. - Philippe Deléham, May 17 2005; corrected by Ilya Gutkovskiy, Jul 25 2018
Ignoring reflections, this is the number of ways of connecting n+2 equally-spaced points on a circle with a path of n+1 line segments. See A030077 for the number of distinct lengths. - T. D. Noe, Jan 05 2007
From Gary W. Adamson, Apr 20 2009: (Start)
Signed: (+ - - + + - - + +, ...) = eigensequence of triangle A002260.
Example: -360 = (1, 1, -1, -4, 12, 71) dot (1, -2, 3, -4, 5, -6) = (1, -2, -3, 16, 60, -432). (End)
a(n) is the number of odd fixed points in all permutations of {1, 2, ..., n+1}, Example: a(2)=4 because we have 1'23', 1'32, 312, 213', 231, and 321, where the odd fixed points are marked. - Emeric Deutsch, Jul 18 2009
a(n) is also the number of permutations of [n+1] starting with an even number. - Olivier Gérard, Nov 07 2011

T. D. Noe, Table of n, a(n) for n=0..100
A. Moreno Canadas, P. F. Fernandez Espinoza, and I. D. M. Gaviria, Categorification of some integer sequences via Kronecker Modules, JP J. Algebra, Number Theory and Applic. 38 (4) (2016) 339-347
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 500.

Cf. A174548, A052841.
Cf. A002260. - Gary W. Adamson, Apr 20 2009
Cf. A052591. - Emeric Deutsch, Jul 18 2009
Cf. A052618, A077611, A199495. - Olivier Gérard, Nov 07 2011

List([0..30], n-> ((-1)^n +2*n +3)*Factorial(n)/4); # G. C. Greubel, May 07 2019

[((-1)^n +2*n +3)*Factorial(n)/4: n in [0..30]]; // G. C. Greubel, May 07 2019

spec := [S,{S=Prod(Sequence(Z),Sequence(Prod(Z,Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Table[n!((-1)^n+2n+3)/4,{n,0,30}] (* Harvey P. Dale, Aug 16 2014 *)

PARI
```
a(n)=if(n<0,0,(1+n\2)*n!)
```

a(n)=if(n<0, 0, n!*polcoeff(1/(1-x)/(1-x^2)+x*O(x^n), n))

[((-1)^n +2*n +3)*factorial(n)/4 for n in (0..30)] # G. C. Greubel, May 07 2019

D-finite with recurrence a(n) = a(n-1) + (n^2-1)*a(n-2), with a(1)=1, a(0)=1.
a(n) = ((-1)^n + 2*n + 3)*n!/4.
Let u(1)=1, u(n) = Sum_{k=1..n-1} u(k)*k*(-1)^(k-1) then a(n) = abs(u(n+2)). - Benoit Cloitre, Nov 14 2003
E.g.f.: 1/((1-x)*(1-x^2)).
From Emeric Deutsch, Jul 18 2009: (Start)
a(n) = (n+1)!/2 if n is odd; a(n) = n!(n+2)/2 if n is even.
a(n) = (n+1)! - A052591(n). (End)
E.g.f.: G(0)/(1+x) where G(k) = 1 + 2*x*(k+1)/((2*k+1) - x*(2*k+1)*(2*k+3)/(x*(2*k+3) + 2*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 21 2012
Sum_{n>=0} 1/a(n) = e - 1/e = 2*sinh(1) (A174548). - Amiram Eldar, Jan 22 2023

A089677 Exponential convolution of A000670(n), with A000670(0)=0, with the sequence of all ones alternating in sign.

Original entry on oeis.org

0, 1, 1, 7, 37, 271, 2341, 23647, 272917, 3543631, 51123781, 811316287, 14045783797, 263429174191, 5320671485221, 115141595488927, 2657827340990677, 65185383514567951, 1692767331628422661, 46400793659664205567, 1338843898122192101557

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Jan 03 2004

Stirling transform of A005212(n)=[1,0,6,0,120,0,5040,...] is a(n)=[1,1,7,37,271,...]. - Michael Somos, Mar 04 2004

Occurs also as first column of a matrix-inversion occurring in a sum-of-like-powers problem. Consider the problem for any fixed natural number m>2 of finding solutions to sum(k=1,n,k^m) = (k+1)^m. Erdos conjectured that there are no solutions for n,m>2. Let D be the matrix of differences of D[m,n] := sum(k=1,n,k^m) - (k+1)^m. Then the generating functions for the rows of this matrix D constitute a set of polynomials in n (for varying n along columns) and the m-th polynomial defining the m-th row. Let GF_D be the matrix of the coefficients of this set of polynomials. Then the present sequence is the (unsigned) second column of GF_D^-1. - Gottfried Helms, Apr 01 2007

			From _Gus Wiseman_, Jan 06 2021: (Start)
a(n) is the number of ordered set partitions of {1..n} into an odd number of blocks. The a(1) = 1 through a(3) = 7 ordered set partitions are:
  {{1}}  {{1,2}}  {{1,2,3}}
                  {{1},{2},{3}}
                  {{1},{3},{2}}
                  {{2},{1},{3}}
                  {{2},{3},{1}}
                  {{3},{1},{2}}
                  {{3},{2},{1}}
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J.-C. Aval, V. Féray, J.-C. Novelli, J.-Y. Thibon, Quasi-symmetric functions as polynomial functions on Young diagrams, arXiv preprint arXiv:1312.2727, 2013
Gottfried Helms, Discussion of a problem concerning summing of like powers

Ordered set partitions are counted by A000670.
The case of (unordered) set partitions is A024429.
The complement (even-length ordered set partitions) is counted by A052841.
A058695 counts partitions of odd numbers, ranked by A300063.
A101707 counts partitions of odd positive rank.
A160786 counts odd-length partitions of odd numbers, ranked by A300272.
A340102 counts odd-length factorizations into odd factors.
A340692 counts partitions of odd rank.
Other cases of odd length:
- A027193 counts partitions of odd length.
- A067659 counts strict partitions of odd length.
- A166444 counts compositions of odd length.
- A174726 counts ordered factorizations of odd length.
- A332304 counts strict compositions of odd length.
- A339890 counts factorizations of odd length.
Cf. A000700, A026424, A027187, A028260, A078408, A174725, A236914, A340101.

h := n -> add(combinat:-eulerian1(n,k)*2^k,k=0..n):
a := n -> (h(n)-(-1)^n)/2: seq(a(n),n=0..20); # Peter Luschny, Jul 09 2015

Table[Sum[Binomial[n, k](-1)^(n-k)Sum[i! StirlingS2[k, i], {i, 1, k}], {k, 0, n}], {n, 0, 20}]

a(n)=if(n<0,0,n!*polcoeff(subst(y/(1-y^2),y,exp(x+x*O(x^n))-1),n))

{a(n)=polcoeff(sum(m=0,n,(2*m+1)!*x^(2*m+1)/prod(k=1,2*m+1,1-k*x+x*O(x^n))),n)} /* Paul D. Hanna, Jul 20 2011 */

def A089677_list(len):  # with a(0)=1
    e, r = [1], [1]
    for i in (1..len-1):
        for k in range(i-1, -1, -1): e[k] = (e[k]*i)//(i-k)
        r.append(-sum(e[j]*(-1)^(i-j) for j in (0..i-1)))
        e.append(sum(e))
    return r
A089677_list(21) # Peter Luschny, Jul 09 2015

E.g.f.: (exp(x)-1)/(exp(x)*(2-exp(x))).
O.g.f.: Sum_{n>=0} (2*n+1)! * x^(2*n+1) / Product_{k=1..2*n+1} (1-k*x). - Paul D. Hanna, Jul 20 2011
a(n)=Sum(Binomial(n, k)(-1)^(n-k)Sum(i! Stirling2(k, i), i=1, ..k), k=0, .., n).
a(n) = (A000670(n)-(-1)^n)/2. - Vladeta Jovovic, Jan 17 2005
a(n) ~ n! / (4*(log(2))^(n+1)). - Vaclav Kotesovec, Feb 25 2014
a(n) = Sum_{k=0..floor(n/2)} (2*k+1)!*Stirling2(n, 2*k+1). - Peter Luschny, Sep 20 2015

A330047 Expansion of e.g.f. exp(-x) / (1 - sinh(x)).

Original entry on oeis.org

1, 0, 1, 3, 13, 75, 511, 4053, 36793, 375735, 4262971, 53203953, 724379173, 10684377795, 169713810631, 2888340723453, 52433443111153, 1011340189494255, 20654264750645491, 445249365444296553, 10103533212012216733, 240731286454287293115, 6008902898851584479551

Offset: 0

Ilya Gutkovskiy, Nov 28 2019

nonn

Inverse binomial transform of A006154.

Seiichi Manyama, Table of n, a(n) for n = 0..440

Cf. A006154, A009227, A052841, A330046, A346894, A346895, A346921.

nmax = 22; CoefficientList[Series[Exp[-x]/(1 - Sinh[x]), {x, 0, nmax}], x] Range[0, nmax]!

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(exp(x)-1)^2/2))) \\ Seiichi Manyama, May 07 2022

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (2*k)!*x^(2*k)/(2^k*prod(j=1, 2*k, 1-j*x)))) \\ Seiichi Manyama, May 07 2022

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, binomial(i, j)*stirling(j, 2, 2)*v[i-j+1])); v; \\ Seiichi Manyama, May 07 2022

a(n) = sum(k=0, n\2, (2*k)!*stirling(n, 2*k, 2)/2^k); \\ Seiichi Manyama, May 07 2022

a(n) = Sum_{k=0..n} (-1)^(n - k) * binomial(n,k) * A006154(k).
a(n) ~ n! / ((2 + sqrt(2)) * (log(1 + sqrt(2)))^(n+1)). - Vaclav Kotesovec, Dec 03 2019
From Seiichi Manyama, May 07 2022: (Start)
E.g.f.: 1/(1 - (exp(x) - 1)^2 / 2).
G.f.: Sum_{k>=0} (2*k)! * x^(2*k)/(2^k * Product_{j=1..2*k} (1 - j * x)).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling2(k,2) * a(n-k).
a(n) = Sum_{k=0..floor(n/2)} (2*k)! * Stirling2(n,2*k)/2^k. (End)
a(0) = 1; a(n) = (-1)^n + Sum_{k=1..ceiling(n/2)} binomial(n,2*k-1) * a(n-2*k+1). - Prabha Sivaramannair, Oct 06 2023

A367977 Expansion of e.g.f. exp(-x) / (2 - exp(2*x)).

Original entry on oeis.org

1, 1, 9, 73, 849, 12241, 211929, 4280473, 98806689, 2565862561, 74035143849, 2349822967273, 81361870604529, 3051889548205681, 123282485663042169, 5335770920836028473, 246332487897909570369, 12083010395805261921601, 627555570373369525058889, 34404109751876393769480073

Offset: 0

Ilya Gutkovskiy, Dec 07 2023

nonn

G. C. Greubel, Table of n, a(n) for n = 0..380

Cf. A000670, A052841, A080253, A216794, A344037, A367979, A367980, A367981, A367982, A367983.

R:=PowerSeriesRing(Rationals(), 50);
Coefficients(R!(Laplace( Exp(-x)/(2-Exp(2*x)) ))) // G. C. Greubel, Jun 10 2024

nmax = 19; CoefficientList[Series[Exp[-x]/(2 - Exp[2 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (-1)^n + Sum[Binomial[n, k] 2^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

def A367977_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-x)/(2-exp(2*x)) ).egf_to_ogf().list()
A367977_list(50) # G. C. Greubel, Jun 10 2024

a(n) = Sum_{k>=0} (2*k-1)^n / 2^(k+1).
a(n) = (-1)^n + Sum_{k=1..n} binomial(n,k) * 2^k * a(n-k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 2^k * A000670(k).

A344037 Expansion of e.g.f.: exp(-2*x) / (2 - exp(x)).

Original entry on oeis.org

1, -1, 3, -1, 27, 119, 1203, 11759, 136587, 1771559, 25562403, 405657119, 7022893947, 131714582999, 2660335750803, 57570797728079, 1328913670528107, 32592691757218439, 846383665814342403, 23200396829831840639, 669421949061096575067, 20281206249626017421879

Offset: 0

Ilya Gutkovskiy, Aug 01 2021

sign

G. C. Greubel, Table of n, a(n) for n = 0..420

Cf. A000670, A007047, A008619, A052841, A330603, A346208.

R:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!(Laplace( Exp(-2*x)/(2-Exp(x)) ))); // G. C. Greubel, Jun 11 2024

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

def A344037_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-2*x)/(2-exp(x)) ).egf_to_ogf().list()
A344037_list(40) # G. C. Greubel, Jun 11 2024

a(n) = Sum_{k=0..n} binomial(n,k) * (-2)^(n-k) * A000670(k).
a(n) = Sum_{k=0..n} (-1)^k * Stirling2(n,k) * k! * A008619(k).
a(n) = Sum_{k>=0} (k - 2)^n / 2^(k+1).
a(n) = (-2)^n + Sum_{k=0..n-1} binomial(n,k) * a(k).
a(n) ~ n! / (8 * log(2)^(n+1)). - Vaclav Kotesovec, Aug 15 2021

cp's OEIS Frontend

A336102 Number of inseparable multisets of size n covering an initial interval of positive integers.

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A349050 Number of multisets of size n that have no alternating permutations and cover an initial interval of positive integers.

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A340602 Heinz numbers of integer partitions of even rank.

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A349055 Number of multisets of size n that have an alternating permutation and cover an initial interval of positive integers.

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A154921 Triangle read by rows, T(n, k) = binomial(n, k) * Sum_{j=0..n-k} E(n-k, j)*2^j, where E(n, k) are the Eulerian numbers A173018(n, k), for 0 <= k <= n.

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A052558 a(n) = n! *((-1)^n + 2*n + 3)/4.

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A052558 a(n) = n! ((-1)^n + 2n + 3)/4.