A098558 -id:A098558 - cp's OEIS frontend

A256881 a(n) = n!/ceiling(n/2).

1, 2, 3, 12, 40, 240, 1260, 10080, 72576, 725760, 6652800, 79833600, 889574400, 12454041600, 163459296000, 2615348736000, 39520825344000, 711374856192000, 12164510040883200, 243290200817664000, 4644631106519040000, 102181884343418880000, 2154334728240414720000

Offset: 1

M. F. Hasler, Apr 22 2015

nonn

Original name was: n!/round(n/2). - Robert Israel, Sep 03 2018

Robert Israel, Table of n, a(n) for n = 1..450
Pierre-Alain Sallard, Sum of repeated integrals of sinh.

Cf. A009445, A052612, A052616, A052849, A081457, A208529, A098558, A107991, A110468, A229244, A256031.

[Factorial(n)/Round(n/2): n in [1..30]]; // Vincenzo Librandi, Apr 23 2015

Maple
```
A256881 := n!/round(n/2);
```

Function[x, 1/x] /@
CoefficientList[Series[(Sinh[x] + x*Exp[x])/2, {x, 0, 20}], x] (* Pierre-Alain Sallard, Dec 15 2018 *)

PARI
```
A256881(n)=n!/round(n/2)
```

a(2n) = 2*A009445(n) = A052612(2n-1) = A052616(2n-1) = A052849(2n-1) = A098558(2n-1) = A081457(3n-1) = A208529(2n+1) = A256031(2n-1).
a(2n+1) = A110468(n) = A107991(2n+2) = A229244(2n+1), n>=0.
From Robert Israel, Sep 03 2018: (Start)
E.g.f.: -(1+1/x)*log(1-x^2).
n*(n+1)*(n+2)*a(n)+(n+2)*a(n+1)-(n+3)*a(n+2)=0. (End)
a(n) = 2/([x^n](sinh(x) + x*exp(x))). - Pierre-Alain Sallard, Dec 15 2018
Sum_{n>=1} 1/a(n) = (3*e-1/e)/4 = (e + sinh(1))/2. - Amiram Eldar, Feb 02 2023

Definition clarified by Robert Israel, Sep 03 2018

A344901 Triangle read by rows: T(n,k) is the number of permutations of length n that have k same elements at the same positions with its inverse permutation for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 2, 0, 0, 4, 6, 8, 0, 0, 10, 24, 30, 40, 0, 0, 26, 160, 144, 180, 160, 0, 0, 76, 1140, 1120, 1008, 840, 700, 0, 0, 232, 8988, 9120, 8960, 5376, 4200, 2912, 0, 0, 764, 80864, 80892, 82080, 53760, 30240, 19656, 12768, 0, 0, 2620, 809856, 808640, 808920, 547200, 336000, 157248, 95760, 55680, 0, 0, 9496

Offset: 0

Mikhail Kurkov, Jun 01 2021

			Triangle T(n,k) begins:
     1;
     0,    1;
     0,    0,    2;
     2,    0,    0,    4;
     6,    8,    0,    0,   10;
    24,   30,   40,    0,    0,   26;
   160,  144,  180,  160,    0,    0, 76;
  1140, 1120, 1008,  840,  700,    0,  0, 232;
  8988, 9120, 8960, 5376, 4200, 2912,  0,   0, 764;
  ...

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

Columns k=0-1 give: A038205, A221145.
Row sums give A000142.
Main diagonal gives A000085.
Cf. A000023, A007318, A052571, A052849, A098558, A137482.

b:= proc(n, t) option remember; `if`(n=0, 1, add(b(n-j, t)*
      binomial(n-1, j-1)*(j-1)!, j=`if`(t=1, 1..min(2, n), 3..n)))
    end:
T:= (n, k)-> binomial(n, k)*b(k, 1)*b(n-k, 0):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Oct 28 2024

b[n_, t_] := b[n, t] = If[n == 0, 1, Sum[b[n-j, t]* Binomial[n-1, j-1]*(j-1)!, {j, If[t == 1, Range @ Min[2, n], Range[3, n]]}]];
T[n_, k_] := Binomial[n, k]*b[k, 1]*b[n-k, 0];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 24 2025, after Alois P. Heinz *)

T(n,k) = binomial(n,k)*A000085(k)*A038205(n-k).
From Alois P. Heinz, Oct 28 2024: (Start)
Sum_{k=0..n} k * T(n,k) = A052849(n) = A098558(n) for n>=2.
Sum_{k=0..n} (n-k) * T(n,k) = A052571(n).
Sum_{k=0..n} (-1)^k * T(n,k) = A000023(n).
T(n,0) + T(n,1) = A137482(n). (End)

A356661 a(n) = n! * Sum_{d|n} 1/d^(n/d - 1).

Original entry on oeis.org

1, 4, 12, 60, 240, 1860, 10080, 95760, 766080, 8210160, 79833600, 1100484000, 12454041600, 188172784800, 2683799838720, 44951306400000, 711374856192000, 13745322470880000, 243290200817664000, 5142812718440517120, 103294640229580800000, 2351280996859354560000

Offset: 1

Seiichi Manyama, Aug 21 2022

nonn

Cf. A087905, A087909, A098558, A356662.

a[n_] := n! * DivisorSum[n, 1/#^(n/# - 1) &]; Array[a, 22] (* Amiram Eldar, Aug 21 2022 *)

PARI
```
a(n) = n!*sumdiv(n, d, 1/d^(n/d-1));
```

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/(1-x^k/k))))

a(p) = 2 * p! for prime p.

E.g.f.: Sum_{k>=1} x^k/(1 - x^k/k).

A356662 a(n) = n! * Sum_{d|n} 1/(d!)^(n/d - 1).

Original entry on oeis.org

1, 4, 12, 60, 240, 1740, 10080, 87360, 735840, 7514640, 79833600, 976686480, 12454041600, 175736040480, 2616448554720, 42011071502400, 711374856192000, 12830610027755520, 243290200817664000, 4870565189425615680, 102182981410948838400, 2249099140674523737600

Offset: 1

Seiichi Manyama, Aug 21 2022

nonn

Cf. A061095, A098558, A356543, A356661.

a[n_] := n! * DivisorSum[n, 1/(#!)^(n/# - 1) &]; Array[a, 22] (* Amiram Eldar, Aug 21 2022 *)

PARI
```
a(n) = n!*sumdiv(n, d, 1/d!^(n/d-1));
```

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/(1-x^k/k!))))

a(p) = 2 * p! for prime p.

E.g.f.: Sum_{k>=1} x^k/(1 - x^k/k!).

A066237 First differences give A052849.

Original entry on oeis.org

1, 3, 7, 19, 67, 307, 1747, 11827, 92467, 818227, 8075827, 87909427, 1045912627, 13499954227, 187856536627, 2803205272627, 44648785048627, 756023641240627, 13560771052696627, 256850971870360627, 5122654988223640627

Offset: 1

Markus Sullivan (markus(AT)o-reading.co.uk), Dec 19 2001

Partial sums of A098558. - Sébastien Desbordes, Dec 18 2023

Sébastien Desbordes, Showing why it is the partial sums of A098558

Cf. A003422, A052849, A098558, A052582.

RecurrenceTable[{a[0]==-1,a[1]==1,a[n]==n*a[n-1]-(n-1)a[n-2]},a,{n,30}] (* Harvey P. Dale, Dec 10 2013 *)

From Vladeta Jovovic, Dec 20 2001: (Start)
a(n) = n*a(n-1) - (n-1)*a(n-2), a(0)=-1, a(1)=1.
a(n) = 2*A003422(n) - 1. (End)

More terms from Jason Earls, Jan 13 2002

A158471 Stirling-like triangle by rows generated from (x-1)(x-1)(x-2)(x-3)(x-4)*...

Original entry on oeis.org

1, 1, -1, 1, -2, 1, 1, -4, 5, -2, 1, -7, 17, -17, 6, 1, -11, 45, -85, 74, -24, 1, -16, 100, -310, 499, -394, 120, 1, -22, 196, -910, 2359, -3388, 2484, -720, 1, -29, 350, -2282, 8729, -19901, 26200, -18108, 5040, 1, -37, 582, -5082, 26985, -89733, 185408

Offset: 0

Gary W. Adamson, Mar 20 2009

Row sums of the unsigned triangle = A098558: (1, 2, 4, 12, 48, 240, 1440, 10080, ...).

			First few rows of the unsigned triangle =
1;
1, 1;
1, 2, 1;
1, 4, 5, 2;
1, 7, 17, 17, 6;
1, 11, 45, 85, 74, 24;
1, 16, 100, 310, 499, 394, 120;
1, 22, 196, 910, 2359, 3388, 2484, 720;
1, 29, 350, 2282, 8729, 19901, 26200, 18108, 5040;
1, 37, 582, 5082, 26985, 89733, 185408, 227708, 149904, 40320;
1, 46, 915, 10320, 72723, 332598, 993005, 1896380, 2199276, 1389456, 362880;
...
Example: Row 5 = x^5 - 11x^4 + 45x^3 -85x^2 + 74x - 24 =
(x-1)*(x-1)*(x-2)*(x-3)*(x-4).

Cf. A098558.

Cf. A103718. - R. J. Mathar, Mar 20 2009

Triangle read by rows, n-th row = n-th degree polynomial with alternating signs generated from n terms of a*b*c*d*...; where a,b,c,... = (x-1), (x-1), (x-2), (x-3), (x-4), ... n-th row, n > 0 = charpoly of an n X n matrix with (1,1,2,3,4,...) in the diagonal and the rest zeros.

A278463 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Original entry on oeis.org

1, 2, 2, 3, 9, 4, 4, 36, 44, 12, 5, 110, 355, 250, 48, 6, 300, 2010, 3480, 1644, 240, 7, 777, 9625, 32235, 35728, 12348, 1440, 8, 1960, 42056, 242200, 498512, 390880, 104544, 10080, 9, 4860, 173754, 1605744, 5466321, 7745220, 4581036, 986256, 80640, 10, 11880, 691620, 9807840, 51506490, 117711720, 123330680, 57537360, 10265760, 725760

Offset: 1

Gheorghe Coserea, Jan 18 2017

			A(x;t) = x + (2*t+2)*x^2/2! + (3*t^2+9*t+4)*x^3/3! + (4*t^3+36*t^2+44*t+12)*x^4/4! + ...
Triangle starts:
n\k  [1]      [2]      [3]      [4]      [5]      [6]      [7]
[1]  1;
[2]  2,       2;
[3]  3,       9,       4;
[4]  4,       36,      44,      12;
[5]  5,       110,     355,     250,     48;
[6]  6,       300,     2010,    3480,    1644,    240;
[7]  7,       777,     9625,    32235,   35728,   12348,   1440;
[8]  ...

Gheorghe Coserea, Rows n = 1..101, flattened.
F. Chapoton, F. Hivert, J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.

N=11; x = 'x+O('x^N);
concat(apply(p->Vec(p), Vec(serlaplace((t-1)*log(1-x) - log(-x + exp(t*log(1-x))) - x))))

A(x;t) = Sum {n>=1} P_n(t)*x^n/n! = (t-1)*log(1-x) - log(-x + exp(t*log(1-x))) - x.
A278458(x;t) = serreverse(A(-x;t))(-x).
A098558(n-1) = P_n(0), A032184(n) = P_n(1).
A052881(n) = T(n,n-1).

A356639 Number of integer sequences b with b(1) = 1, b(m) > 0 and b(m+1) - b(m) > 0, of length n which transform under the map S into a nonnegative integer sequence. The transform c = S(b) is defined by c(m) = Product_{k=1..m} b(k) / Product_{k=2..m} (b(k) - b(k-1)).

Original entry on oeis.org

1, 1, 3, 17, 155, 2677, 73327, 3578339, 329652351

Offset: 1

Thomas Scheuerle, Aug 19 2022

This sequence can be calculated by a recursive algorithm:
Let B1 be an array of finite length, the "1" denotes that it is the first generation. Let B1' be the reversed version of B1. Let C be the element-wise product C = B1 * B1'. Then B2 is a concatenation of taking each element of B1 and add all divisors of the corresponding element in C. If we start with B1 = {1} then we get this sequence of arrays: B2 = {2}, B3 = {3, 4, 6}, ... . a(n) is the length of the array Bn. In short the length of Bn+1 and so a(n+1) is the sum over A000005(Bn * Bn').
The transform used in the definition of this sequence is its own inverse, so if c = S(b) then b = S(c). The eigensequence is 2^n = S(2^n).
There exist some transformation pairs of infinite sequences in the database:
  A000027 <--> A000142; A000079 <--> A000079; A000045 <--> A001654;
  A026549 <--> A038754; A100071 <--> A001405; A058295 <--> A------;
  A111286 <--> A098011; A093968 <--> A205825; A166447 <--> A------;
  A098558 <--> A004277; A010551 <--> A087811; A100538 <--> A329227;
  A079352 <--> A------; A082458 <--> A------; A008233 <--> A264635;
  A138278 <--> A------; A006501 <--> A264557; A336496 <--> A------;
  A019464 <--> A------; A062112 <--> A------; A171647 <--> A359039;
  A279312 <--> A------; A031923 <--> A------.
These transformation pairs are conjectured:
  A137326 <--> A------; A066332 <--> A300902; A208147 <--> A308546;
  A057895 <--> A------; A349080 <--> A------; A019442 <--> A------;
  A349079 <--> A------.
("A------" means not yet in the database.)
Some sequences in the lists above may need offset adjustment to force a beginning with 1,2,... in the transformation.
If we allowed signed rational numbers, further interesting transformation pairs could be observed. For example, 1/n will transform into factorials with alternating sign. 2^(-n) transforms into ones with alternating sign and 1/A000045(n) into A000045 with alternating sign.

			a(4) = 17. The 17 transformation pairs of length 4 are:
  {1, 2, 3, 4}  = S({1, 2, 6, 24}).
  {1, 2, 3, 5}  = S({1, 2, 6, 15}).
  {1, 2, 3, 6}  = S({1, 2, 6, 12}).
  {1, 2, 3, 9}  = S({1, 2, 6, 9}).
  {1, 2, 3, 12} = S({1, 2, 6, 8}).
  {1, 2, 3, 21} = S({1, 2, 6, 7}).
  {1, 2, 4, 5}  = S({1, 2, 4, 20}).
  {1, 2, 4, 6}  = S({1, 2, 4, 12}).
  {1, 2, 4, 8}  = S({1, 2, 4, 8}).
  {1, 2, 4, 12} = S({1, 2, 4, 6}).
  {1, 2, 4, 20} = S({1, 2, 4, 5}).
  {1, 2, 6, 7}  = S({1, 2, 3, 21}).
  {1, 2, 6, 8}  = S({1, 2, 3, 12}).
  {1, 2, 6, 9}  = S({1, 2, 3, 9}).
  {1, 2, 6, 12} = S({1, 2, 3, 6}).
  {1, 2, 6, 15} = S({1, 2, 3, 5}).
  {1, 2, 6, 24} = S({1, 2, 3, 4}).
b(1) = 1 by definition, b(2) = 1+1 as 1 has only 1 as divisor.
a(3) = A000005(b(2)*b(2)) = 3.
The divisors of b(2) are 1,2,4. So b(3) can be b(2)+1, b(2)+2 and b(2)+4.
a(4) = A000005((b(2)+1)*(b(2)+4)) + A000005((b(2)+2)*(b(2)+2)) + A000005((b(2)+4)*(b(2)+1)) = 17.

Cf. A000005.
Cf. A000027, A000045, A000079, A000142, A001405.
Cf. A001654, A004277, A006501, A008233, A019442.
Cf. A010551, A019464, A026549, A031923, A038754.
Cf. A057895, A058295, A062112, A066332, A100071.
Cf. A079352, A082458, A087811, A093968, A098011.
Cf. A098558, A100538, A111286, A166447, A137326.
Cf. A138278, A171647, A205825, A208147, A264557.
Cf. A264635, A308546, A329227, A336496, A349079.
Cf. A349080, A359039.

A142156 Triangle T(n,k)= n! if k=0, T(n,k) = -(n-k)!*A003319(k) if k > 0.

Original entry on oeis.org

1, 1, -1, 2, -1, -1, 6, -2, -1, -3, 24, -6, -2, -3, -13, 120, -24, -6, -6, -13, -71, 720, -120, -24, -18, -26, -71, -461, 5040, -720, -120, -72, -78, -142, -461, -3447, 40320, -5040, -720, -360, -312, -426, -922, -3447, -29093, 362880, -40320, -5040, -2160, -1560

Offset: 0

Paul Curtz, Sep 16 2008

			Triangle begins
    1;
    1,   -1;
    2,   -1,  -1;
    6,   -2,  -1,  -3;
   24,   -6,  -2,  -3, -13;
  120,  -24,  -6,  -6, -13, -71;
  720, -120, -24, -18, -26, -71, -461;

(* b = A003319 *) b[0]=0; b[n_] := b[n] = n! - Sum[k! b[n-k], {k, 1, n-1}]; T[n_, 0] := n!; T[n_, k_] := -(n - k)! b[k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 18 2018 *)

T(n,0) = A000142(n).
T(n,k) = -A141476(n-1,k-1), k > 0.
Sum_{k=0..n} |T(n,k)| = 2*n! = A098558(n).

A198895 Triangle of coefficients arising in expansion of n-th derivative of tan(x) + sec(x).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 5, 2, 1, 8, 18, 16, 5, 1, 16, 58, 88, 61, 16, 1, 32, 179, 416, 479, 272, 61, 1, 64, 543, 1824, 3111, 2880, 1385, 272, 1, 128, 1636, 7680, 18270, 24576, 19028, 7936, 1385, 1, 256, 4916, 31616, 101166, 185856, 206276

Offset: 0

N. J. A. Sloane, Oct 31 2011

From Petros Hadjicostas, Aug 10 2019: (Start)
The recurrence about T(n, k) and the equation that connects T(n, k) to P(n, k) = A059427(n,k), which are given below, appear on p. 159 of the book by David and Barton (1962). The initial conditions, however, for their triangular array S^*{N,t} are slightly different, but there is an agreement starting at t = k = 1. They do not provide tables for S^*{N,t} (that matches the current array T(n, k) for N = n >= 0 and t = k >= 1).
Despite the slightly different initial conditions between T(n, k) and S^*_{N,t} (from p. 159 in the book), the recurrence given below can be proved very easily from the recurrence for the row polynomials R_n(x) given in Shi-Mei Ma (2011, 2012). (End)

			Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
  1
  1   1
  1   2    1
  1   4    5     2
  1   8   18    16      5
  1  16   58    88     61     16
  1  32  179   416    479    272     61
  1  64  543  1824   3111   2880   1385    272
  1 128 1636  7680  18270  24576  19028   7936  1385
  1 256 4916 31616 101166 185856 206276 137216 50521 7936
  ...

Florence Nightingale David and D. E. Barton, Combinatorial Chance, Charles Griffin, 1962; see pp. 159-162.

Shi-Mei Ma, Derivative polynomials and permutations by numbers of interior peaks and left peaks, arXiv:1106.5781 [math.CO], 2011.
Shi-Mei Ma, Derivative polynomials and enumeration of permutations by number of interior and left peaks , Discrete Mathematics 312(2) (2012), 405-412.
Wikipedia, Florence Nightingale David.

Cf. A059427, A098558 (row sums), A000111 (diagonal and 1st subdiagonal), A000340 (column 3) A000431 (column 4), A000363 (column 5)

n-th row represents the coefficients of the polynomial R_n(x) defined by the recurrence: R_0(x) = 1, R_1(x) = 1 + x, and for n >= 1, R_{n+1}(x) = (1 + n*x^2)*R_n(x) + x*(1 - x^2)*R'_n(x).
From Petros Hadjicostas, Aug 10 2019: (Start)
T(n, k) = (k + 1) * T(n-1, k) + (n - k + 1) * T(n-1, k-2) for n >= 0 and 2 <= k <= n with initial conditions T(n, k=0) = 1 for n >= 0, T(n, k=1) = 2^(n-1) for n >= 1, and T(n, k) = 0 for n < 0 or n < k.
Setting x = 1 in the equation R_{n+1}(x) = (1 + n*x^2)*R_n(x) + x*(1 - x^2)*R'n(x) (valid for n >= 1), we get R{n+1}(1) = (n + 1)*R_n(1) for n >= 1. Since R_1(1) = 2, we have that R_n(1) = 2*n! for n >= 1. Since also R_0(1) = 1, we conclude that Sum_{k = 0..n} T(n,k) = R_n(1) = 2*n! - 0^n = A098558(n) for n >= 0.
Let P(n, k) = A059427(n,k) with P(n, k) = 0 for n <= 1 or n <= k. Then T(n, k) = (1/2)*P(n, k-1) + P(n, k) + (1/2) * P(n, k+1) for n >= 2 and 0 <= k <= n (but this is not true for n = 0 and n = 1). (End)

More terms from Max Alekseyev, Feb 17 2012

cp's OEIS Frontend

A256881 a(n) = n!/ceiling(n/2).

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A344901 Triangle read by rows: T(n,k) is the number of permutations of length n that have k same elements at the same positions with its inverse permutation for 0 <= k <= n.

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A356661 a(n) = n! * Sum_{d|n} 1/d^(n/d - 1).

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A356662 a(n) = n! * Sum_{d|n} 1/(d!)^(n/d - 1).

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A066237 First differences give A052849.

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A158471 Stirling-like triangle by rows generated from (x-1)*(x-1)*(x-2)*(x-3)*(x-4)*...

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A278463 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

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A356639 Number of integer sequences b with b(1) = 1, b(m) > 0 and b(m+1) - b(m) > 0, of length n which transform under the map S into a nonnegative integer sequence. The transform c = S(b) is defined by c(m) = Product_{k=1..m} b(k) / Product_{k=2..m} (b(k) - b(k-1)).

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A158471 Stirling-like triangle by rows generated from (x-1)(x-1)(x-2)(x-3)(x-4)*...