A000111 -id:A000111 - cp's OEIS frontend

A245377 Number of 2-alternating permutations of 1,2,...,n, that is, a(n) is the number of down/up permutations (A000111) of 1,2,...,n such that any two consecutive terms differ by at least two.

1, 1, 0, 0, 1, 4, 17, 80, 422, 2480, 16095, 114432, 884969, 7398464, 66502048, 639653632, 6556170841, 71340409600, 821408397105, 9977630263296, 127518757153174, 1710576547456000, 24030971882538671, 352843606806499328

Offset: 0

Richard Stanley, Jul 19 2014

			For n=5 there are the four permutations 31425, 31524, 52413, 42513.

Cf. A002464.

b:= proc(n, s, t) option remember; `if`(s={}, 1, add(
     `if`(t*(n-j)>=2, b(j, s minus{j}, -t), 0), j=s))
    end:
a:= n->`if`(n=0, 1, add(b(j, {$1..j-1, $j+1..n}, 1), j=1..n)):
seq(a(n), n=0..16);  # Alois P. Heinz, Oct 27 2014

b[n_, s_, t_] := b[n, s, t] = If[s == {}, 1, Sum[If[t*(n - j) >= 2, b[j, s ~Complement~ {j}, -t], 0], {j, s}]]; a[n_] := a[n] = If[n == 0, 1, Sum[b[j, DeleteCases[Range[n], j], 1], {j, 1, n}]]; Table[Print[a[n]]; a[n], {n, 0, 16}] (* Jean-François Alcover, Oct 24 2016, after Alois P. Heinz *)

a(11)-a(15) from R. J. Mathar, Oct 27 2014
a(16)-a(21) from Alois P. Heinz, Oct 27 2014
a(22) from Alois P. Heinz, Feb 18 2024
a(23) from Max Alekseyev, Feb 19 2024

A350970 Triangle T(n,k) (n>=0, 0<=k<=n) read by rows: T(0,0)=T(1,1)=1; T(n,0) is the Euler number A000111(n-1) for n>=1; T(n,n-1) = T(n,n) = (n-2)! for n>=2; interior entries are given by T(n,k) = mT(n-1,k-1)+(k+1)T(n-1,k+1) where m = k if n+k is even or k-1 if n+k is odd.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 5, 8, 6, 6, 5, 16, 28, 40, 24, 24, 16, 61, 136, 180, 240, 120, 120, 61, 272, 662, 1232, 1320, 1680, 720, 720, 272, 1385, 3968, 7266, 12096, 10920, 13440, 5040, 5040, 1385, 7936, 24568, 56320, 83664, 129024, 100800, 120960, 40320, 40320, 7936, 50521, 176896, 408360, 814080, 1023120, 1491840, 1028160, 1209600, 362880, 362880

Offset: 0

N. J. A. Sloane, Mar 03 2022

Triangle connects Euler numbers on left and factorial numbers on right.

			Triangle begins:
    1,
    1,    1,
    1,    1,    1,
    1,    2,    2,    2,
    2,    5,    8,    6,     6,
    5,   16,   28,   40,    24,    24,
   16,   61,  136,  180,   240,   120,   120,
   61,  272,  662, 1232,  1320,  1680,   720,  720,
  272, 1385, 3968, 7266, 12096, 10920, 13440, 5040, 5040,
  ...
This may also be constructed as a square array, with entries T(n,k), n >= 1, 0 <= k, whose columns have e.g.f. equal to sec(x)+tan(x) (if k=0) and sec(x)*tan(x)^(k-1)*(sec(x)+tan(x)) (if k>0):
1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
2, 5, 8, 6, 6, 0, 0, 0, 0, 0, 0, 0, 0, ...
5, 16, 28, 40, 24, 24, 0, 0, 0, 0, 0, 0, 0, ...
16, 61, 136, 180, 240, 120, 120, 0, 0, 0, 0, 0, 0, ...
61, 272, 662, 1232, 1320, 1680, 720, 720, 0, 0, 0, 0, 0, ...
272, 1385, 3968, 7266, 12096, 10920, 13440, 5040, 5040, 0, 0, 0, 0, ...
...

A. Boutin, Query 2784, L'Intermédiaire des Mathématiciens, 11 (1904), 252-254.
E. Estanave, Query 2784, L'Intermédiaire des Mathématiciens, 11 (1904), pp. 117-118.

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

The initial columns are A000111, A000111, A225689, A350971.
The diagonals, reading from the right, are (essentially) A000142, A000142, A002301, A006157, A002302, A350973, A002303, A350974, A350975.
Rows sums give A156142(n-1).
Cf. A007836.

for n from 0 to 12 do
T[n]:=Array(0..n,0);
T[0,0] := 1;
T[1,0] := 1; T[1,1] := 1;
if n>1 then
  T[n,0] := T[n-1,1];
for k from 1 to n-2 do
m:=k; if ((n+k) mod 2) = 0 then m:=k-1; fi;
T[n,k] := m*T[n-1,k-1] + (k+1)*T[n-1,k+1];
od:
T[n,n-1] := (n-1)*T[n-1,n-2];
T[n,n] := T[n,n-1];
fi;
lprint( [seq(T[n,k],k=0..n)] );
od:
# second Maple program:
b:= proc(n, i) option remember; `if`(i=0,
     `if`(n=0, 1, 0), b(n, i-1)+b(n-1, n-i))
    end:
T:= proc(n, k) option remember; `if`(n=0 and k=0, 1,
     `if`(k=0, b(n-1$2), `if`(n-k<=1, (n-1)!, (k+1)*
      T(n-1, k+1)+(k-irem(1+n+k, 2))*T(n-1, k-1))))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Mar 04 2022
# To produce the square array, N. J. A. Sloane, Mar 05 2022:
read(transforms):
myegf := (f,M) -> SERIESTOLISTMULT(series(f,x,M));
T:=proc(n,k,M) local i;
if k=0 then myegf((sec(x)+tan(x)),M)[n];
else
myegf(sec(x)*tan(x)^(k-1)*(sec(x)+tan(x)),M)[n];
fi;
end;
[seq(T(n,0,16),n=1..5)];
for n from 1 to 8 do
lprint([seq(T(n,k,16),k=0..12)]);
od:

b[n_, i_] := b[n, i] = If[i == 0,
     If[n == 0, 1, 0], b[n, i - 1] + b[n - 1, n - i]];
T[n_, k_] := T[n, k] = If[n == 0 && k == 0, 1,
     If[k == 0, b[n - 1, n - 1], If[n - k <= 1, (n - 1)!, (k + 1)*
     T[n - 1, k + 1] + (k - Mod[1 + n + k, 2])*T[n - 1, k - 1]]]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 12 2022, Alois P. Heinz *)

If we ignore the n=0 row, then the e.g.f. for column 0 is sec(x)+tan(x), and for column k >= 1 it is sec(x)*tan(x)^(k-1)*(sec(x)+tan(x)). See the initial rows of the square array in the EXAMPLES section. - N. J. A. Sloane, Mar 05 2022

abs(Sum_{k=0..n} (-1)^k * T(n,k)) = A007836(n) for n>=2. - Alois P. Heinz, Mar 04 2022

A260786 Twice the Euler or up/down numbers A000111.

Original entry on oeis.org

2, 2, 2, 4, 10, 32, 122, 544, 2770, 15872, 101042, 707584, 5405530, 44736512, 398721962, 3807514624, 38783024290, 419730685952, 4809759350882, 58177770225664, 740742376475050, 9902996106248192, 138697748786275802, 2030847773013704704, 31029068327114173810, 493842960380415967232

Offset: 0

N. J. A. Sloane, Aug 04 2015

nonn

S. T. Thompson, Problem E754: Skew Ordered Sequences, Amer. Math. Monthly, 54 (1947), 416-417. [Annotated scanned copy]

Cf. A000111.

Apart from initial terms, same as A001250.

f:=proc(n) option remember;
if n <= 1 then 2 else (1/4)*add(binomial(n-1,k-1)*f(k-1)*f(n-k),k=1..n); fi;
end;
[seq(f(n),n=0..30)];

from itertools import accumulate, islice
def A260786_gen(): # generator of terms
    yield from (2,2)
    blist = (0,2)
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=0)))[-1]
A260786_list = list(islice(A260786_gen(),30)) # Chai Wah Wu, Apr 17 2023

a(0)=a(1)=2; thereafter a(n) = (1/4)*Sum_{k=1..n} binomial(n-1, k-1)*a(k-1)*a(n-k).

A141430 a(n) = A000111(n) mod 9.

Original entry on oeis.org

1, 1, 1, 2, 5, 7, 7, 2, 8, 7, 4, 2, 2, 7, 1, 2, 5, 7, 7, 2, 8, 7, 4, 2, 2, 7, 1, 2, 5, 7, 7, 2, 8, 7, 4, 2, 2, 7, 1, 2, 5, 7, 7, 2, 8, 7, 4, 2, 2, 7

Offset: 0

Paul Curtz, Aug 06 2008

After the initial 1,1, the sequence is periodic with period 12.

This sequence's periodic part is a shuffled version of the two period-6 sequences A070366 and A010697. The sequence contains only the digits 1, 2, 4, 5, 7 and 8 (those of A141425).

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,-1,1).

Cf. A000111, A004099, A010686, A014021.

def A141430(n): return (2, 7, 1, 2, 5, 7, 7, 2, 8, 7, 4, 2)[n%12] if n>1 else 1 # Chai Wah Wu, Apr 17 2023

a(n) = A000111(n) mod 9 = A004099(n) mod 9.
a(n+12) = a(n), n > 1.
a(n) + a(n+6) = 9, n > 1.
a(n+11-p) - a(n+p) = 6 (p=0 or 5), 0 (p=1 or 4), -3 (p=2 or 3), any n > 1.
G.f.: (6x^8-5x^7+x^6+2x^5+3x^4+x^3+1) / ((1-x)(x^2+1)(x^4-x^2+1)). - R. J. Mathar, Dec 05 2008
a(n) = 9/2 +(-1)^floor(n/2)*A010686(n)/2 - 3*A014021(n), n > 1. - R. J. Mathar, Dec 05 2008
a(n) = 9/2 - (3/2)*cos(Pi*n/6) + (1/2)*3^(1/2)*sin(Pi*n/6) - (1/2)*cos(Pi*n/2) - (5/2)*sin(Pi*n/2) - (3/2)*cos(5*Pi*n/6) - (1/2)*3^(1/2)*sin(5*Pi*n/6). - Richard Choulet, Dec 12 2008

Edited by R. J. Mathar, Dec 05 2008

A180418 a(n) = ( A000111(p) -(-1)^((p-1)/2) )/p, where p = prime(n).

Original entry on oeis.org

1, 3, 39, 32163, 1720635, 12345020175, 1530993953307, 44148864630732711, 797213247855503373843915, 281095572810489332134542303, 26242778669866462496740532647355475

Offset: 2

Vladimir Shevelev, Sep 03 2010

nonn

Amiram Eldar, Table of n, a(n) for n = 2..92

Cf. A000040, A000111, A070750.

t = Range[0, 60]! CoefficientList[Series[Sec@x + Tan@x, {x, 0, 60}], x]; f[n_] := (Rest[t][[Prime@n]] - (-1)^((Prime@n - 1)/2))/Prime@n; Array[f, 11, 2] (* Robert G. Wilson v, Sep 04 2010 *)

a(n) = ( A000111(A000040(n)) -A070750(n) )/A000040(n). - R. J. Mathar, Sep 19 2010

Extended by R. J. Mathar, Sep 19 2010

a(8) onwards from Robert G. Wilson v, Sep 04 2010

A373433 a(n) = A000111(n) * A000142(n). Row sums of A373434.

Original entry on oeis.org

1, 1, 2, 12, 120, 1920, 43920, 1370880, 55843200, 2879815680, 183330604800, 14122244505600, 1294628759424000, 139287595371724800, 17379949655535667200, 2489494639794978816000, 405724534220435189760000, 74646464089618378653696000, 15396938399483145082626048000

Offset: 0

Peter Luschny, Jun 04 2024

nonn

Cf. A000111, A000142, A373434.

A373433 := n -> ifelse(n = 0, 1, n! * 2^n * abs(euler(n, 1/2) + euler(n, 1))):
seq(A373433(n), n = 0..18);

A373433[n_] := 2 I^(n + 1) n! PolyLog[-n, -I]; A373433[0] := 1;
Table[A373433[n], {n, 0, 18}]

# Algorithm of Ludwig Seidel (1877).
def A373433_list(n) :
    R = []; S = []; A = {-1:0, 0:1}; k = 0; f = 1; e = 1
    for i in (0..n) :
        Am = 0; A[k + e] = 0; e = -e
        for j in (0..i) : Am += A[k]; A[k] = Am; k += e
        R.append(Am); S.append(f*Am); f *= i + 1
    return S
print(A373433_list(18))

a(n) = n! * 2^n * |Euler(n, 1/2) + Euler(n, 1)| for n >= 1.

a(n) ~ ((2*n^2)/(Pi*e^2))^n*(8*n + 4/3).

A141479 a(n) = A000111(n) + A014695(n).

Original entry on oeis.org

2, 3, 3, 3, 6, 18, 63, 273, 1386, 7938, 50523, 353793, 2702766, 22368258, 199360983, 1903757313, 19391512146, 209865342978, 2404879675443, 29088885112833, 370371188237526, 4951498053124098, 69348874393137903, 1015423886506852353

Offset: 0

Paul Curtz, Aug 09 2008

a(n) is a multiple of 3 for n > 0.

A000111 := proc(n) local x; n!*coeftayl( sec(x)+tan(x),x=0,n) ; end: A014695 := proc(n) local x; coeftayl( (1+2*x+2*x^2+x^3)/(1-x^4),x=0,n) ; end: A141479 := proc(n) A000111(n)+A014695(n) ; end: for n from 0 to 30 do printf("%a,",A141479(n)) ; od; # R. J. Mathar, Sep 12 2008

from itertools import count, islice, accumulate
def A141479_gen(): # generator of terms
    yield from (2,3)
    blist = (0,1)
    for n in count(0):
        yield (blist := tuple(accumulate(reversed(blist),initial=0)))[-1] + (2,1,1,2)[n & 3]
A141479_list = list(islice(A141479_gen(),40)) # Chai Wah Wu, Jun 09-11 2022

Extended by R. J. Mathar, Sep 12 2008

A173253 Partial sums of A000111.

Original entry on oeis.org

1, 2, 3, 5, 10, 26, 87, 359, 1744, 9680, 60201, 413993, 3116758, 25485014, 224845995, 2128603307, 21520115452, 231385458428, 2636265133869, 31725150246701, 402096338484226, 5353594391608322, 74702468784746223, 1090126355291598575, 16604660518848685480

Offset: 0

Jonathan Vos Post, Feb 14 2010

nonn

Partial sums of Euler or up/down numbers. Partial sums of expansion of sec x + tan x. Partial sums of number of alternating permutations on n letters.

			a(22) = 1 + 1 + 1 + 2 + 5 + 16 + 61 + 272 + 1385 + 7936 + 50521 + 353792 + 2702765 + 22368256 + 199360981 + 1903757312 + 19391512145 + 209865342976 + 2404879675441 + 29088885112832 + 370371188237525 + 4951498053124096 + 69348874393137901.

Alois P. Heinz, Table of n, a(n) for n = 0..485

Cf. A000111, A000364, A000182, A008280, A008281, A008282, A010094, A059720, A008970, A109449, A162170.

b:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))
    end:
a:= proc(n) option remember;
      `if`(n<0, 0, a(n-1))+ b(n, 0)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 27 2017

With[{nn=30},Accumulate[CoefficientList[Series[Sec[x]+Tan[x],{x,0,nn}],x] Range[0,nn]!]] (* Harvey P. Dale, Feb 26 2012 *)

from itertools import accumulate
def A173253(n):
    if n<=1:
        return n+1
    c, blist = 2, (0,1)
    for _ in range(n-1):
        c += (blist := tuple(accumulate(reversed(blist),initial=0)))[-1]
    return c # Chai Wah Wu, Apr 16 2023

a(n) = SUM[i=0..n] A000111(i) = SUM[i=0..n] (2^i|E(i,1/2)+E(i,1)| where E(n,x) are the Euler polynomials).
G.f.: (1 + x/Q(0))/(1-x),m=+4,u=x/2, where Q(k) = 1 - 2*u*(2*k+1) - m*u^2*(k+1)*(2*k+1)/( 1 - 2*u*(2*k+2) - m*u^2*(k+1)*(2*k+3)/Q(k+1) ) ; (continued fraction). - Sergei N. Gladkovskii, Sep 24 2013
G.f.: 1/(1-x) + T(0)*x/(1-x)^2, where T(k) = 1 - x^2*(k+1)*(k+2)/(x^2*(k+1)*(k+2) - 2*(1-x*(k+1))*(1-x*(k+2))/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 20 2013
a(n) ~ 2^(n+2)*n!/Pi^(n+1). - Vaclav Kotesovec, Oct 27 2016

A180417 a(n) = (A000111(2^n) - 1) / 2^n.

Original entry on oeis.org

0, 0, 1, 173, 1211969509, 5547480986860602794895774677, 708720364531529518355420122993246286974247836241724513772950684967495246261

Offset: 0

Vladimir Shevelev, Sep 03 2010

nonn

The definition yields integers according to the formula in A000111.

The next term a(7) is too large to include. - N. J. A. Sloane, Mar 14 2011

Alois P. Heinz, Table of n, a(n) for n = 0..8

Cf. A000079, A000111.

t = Range[0, 65]! CoefficientList[ Series[ Sec@x + Tan@x, {x, 0, 65}], x]; f[n_] := (Rest[t][[2^n]] - 1)/2^n; Array[f, 7, 0] (* Robert G. Wilson v, Sep 04 2010 *)

from sympy import euler
def A180417(n): return euler(1<>n if n > 1 else 0 # Chai Wah Wu, Apr 18 2023

Two more terms from R. J. Mathar, Sep 19 2010

a(5) and a(6) from Robert G. Wilson v, Sep 04 2010

A180942 Odd composite numbers m for which A000111(m) == (-1)^( (m-1)/2 ) (mod m).

Original entry on oeis.org

91, 561, 781, 1105, 1661, 1729, 2465, 2737, 2821, 6601, 8911, 10585, 15841, 29341, 30433, 41041, 46657, 52633, 62745, 63973, 75361, 90241, 101101, 115921, 126217, 136371, 136741, 137149, 162401, 172081, 176565, 188461, 251251, 252601, 278545, 294409, 314821, 334153

Offset: 1

Vladimir Shevelev, Sep 27 2010

nonn

For any odd prime p, A000111(p) == (-1)^((p-1)/2) mod p, see A180418, so these cases are not considered further and left out of the sequence by definition.
Might be called "Zig-zag pseudoprimes."
It seems that every Carmichael number (A002997) <= 512461 is in the sequence. - D. S. McNeil, Sep 01 2010

Vladimir Shevelev, The number of permutations with prescribed up-down structure as a function of two variables, INTEGERS 12 (2012), article #A1. - From _N. J. A. Sloane_, Feb 07 2013

Cf. A000111.

fQ[n_] := ! PrimeQ@n && Mod[ (-1)^((n - 1)/2)*2^(n + 1)*(2^(n + 1) - 1)*BernoulliB[n + 1]/(n + 1), n] == Mod[(-1)^((n - 1)/2), n]; k = 3; lst = {}; While[k < 50000, If[ fQ@k, AppendTo[lst, k]; Print@k]; k += 2]; lst (* Robert G. Wilson v, Sep 29 2010 *)

from itertools import count, islice, accumulate
from sympy import isprime
def A180942_gen(): # generator of terms
    blist = (0,1)
    for n in count(2):
        blist = tuple(accumulate(reversed(blist),initial=0))
        if n & 1 and (blist[-1] + (1 if (n-1)//2 & 1 else -1)) % n == 0 and not isprime(n):
            yield n
A180942_list = list(islice(A180942_gen(),5)) # Chai Wah Wu, Jun 09-11 2022

Extended to a(13) by D. S. McNeil, Sep 01 2010
Comments rephrased by R. J. Mathar, Sep 29 2010
a(14)-a(17) from Robert G. Wilson v, Sep 29 2010
a(18)-a(38) from Amiram Eldar, Dec 28 2019

cp's OEIS Frontend

A245377 Number of 2-alternating permutations of 1,2,...,n, that is, a(n) is the number of down/up permutations (A000111) of 1,2,...,n such that any two consecutive terms differ by at least two.

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A350970 Triangle T(n,k) (n>=0, 0<=k<=n) read by rows: T(0,0)=T(1,1)=1; T(n,0) is the Euler number A000111(n-1) for n>=1; T(n,n-1) = T(n,n) = (n-2)! for n>=2; interior entries are given by T(n,k) = m*T(n-1,k-1)+(k+1)*T(n-1,k+1) where m = k if n+k is even or k-1 if n+k is odd.

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A260786 Twice the Euler or up/down numbers A000111.

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A141430 a(n) = A000111(n) mod 9.

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A180418 a(n) = ( A000111(p) -(-1)^((p-1)/2) )/p, where p = prime(n).

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A373433 a(n) = A000111(n) * A000142(n). Row sums of A373434.

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A141479 a(n) = A000111(n) + A014695(n).

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A173253 Partial sums of A000111.

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A350970 Triangle T(n,k) (n>=0, 0<=k<=n) read by rows: T(0,0)=T(1,1)=1; T(n,0) is the Euler number A000111(n-1) for n>=1; T(n,n-1) = T(n,n) = (n-2)! for n>=2; interior entries are given by T(n,k) = mT(n-1,k-1)+(k+1)T(n-1,k+1) where m = k if n+k is even or k-1 if n+k is odd.