A186370 -id:A186370 - cp's OEIS frontend

A240559 a(n) = -2^n(E(n, 1/2) + E(n, 1) + (n mod 2)2*(E(n+1, 1/2) + E(n+1, 1))), where E(n, x) are the Euler polynomials.

0, 0, 1, -3, -5, 45, 61, -1113, -1385, 42585, 50521, -2348973, -2702765, 176992725, 199360981, -17487754833, -19391512145, 2195014332465, 2404879675441, -341282303124693, -370371188237525, 64397376340013805, 69348874393137901, -14499110277050234553

Offset: 0

Peter Luschny, Apr 17 2014

sign

			G.f. = x^2 - 3*x^3 - 5*x^4 + 45*x^5 + 61*x^6 - 1113*x^7 - 1385*x^8 + ...

L. Seidel, Über eine einfache Entstehungsweise der Bernoullischen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, Vol. 7 (1877), 157-187.

Cf. A000364, A147315, A186370, A241242.

Cf. A239005, A239322.

A240559 := proc(n) euler(n,1/2) + euler(n,1); if n mod 2 = 1 then % + 2*(euler(n+1,1/2)+euler(n+1,1)) fi; -2^n*% end: seq(A240559(n),n=0..19);

skp[n_, x_] := Sum[Binomial[n, k]*EulerE[k]*x^(n-k), {k, 0, n}]; skp[n_, x0_?NumericQ] := skp[n, x] /. x -> x0; a[n_] := Sum[(-1)^(n-k)*Binomial[n, k]*(skp[k, 0] + skp[k+1, -1]), {k, 0, n}]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Dec 09 2014, after Peter Luschny *)

# Efficient computation with L. Seidel's boustrophedon transformation.
def A240559_list(n) :
    A = [0]*(n+1); A[0] = 1; R = [0]
    k = 0; e = 1; x = -1; s = -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
        if e == 1: x += 1; s = -s
        v = -A[-x] if e == 1 else A[-x] - A[x]
        if i > 1: R.append(s*v)
    return R
A240559_list(24)

a(2*n) = (-1)^(n+1)*A147315(2*n,1) = (-1)^(n+1)*A186370(2*n,2*n) =(-1)^(n+1)*A000364(n) for n>0.
a(2*n+1) = (-1)^n*A147315(2*n+1,2) = (-1)^n*A186370(2*n,2*n-1) = A241242(n).
a(n) = Sum_{k=0..n} (-1)^(n-k)*2^k*binomial(n,k)*(E(k,1/2) + 2*E(k+1,0)) where E(n,x) are the Euler polynomials.
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*(skp(k,0) + skp(k+1,-1)), where skp(n, x) are the Swiss-Knife polynomials A153641.
a(n) = A239322(n) + A239005(n+1) - A239005(n). - Paul Curtz, Apr 18 2014
E.g.f.: 1 - sech(x) - tanh(x) + sinh(x)*sech(x)^2 = ((exp(-x)-1)*sech(x))^2 / 2. - Sergei N. Gladkovskii, Nov 20 2014
E.g.f.: (1 - sech(x)) * (1 - tanh(x)). - Michael Somos, Nov 22 2014

A291677 Number of permutations p of [2n] such that 0p has exactly n alternating runs.

Original entry on oeis.org

1, 1, 7, 148, 6171, 425976, 43979902, 6346283560, 1219725741715, 301190499710320, 92921064554444490, 35025128774218944648, 15838288022236083603486, 8462453158197423495502224, 5274234568391796228927038748, 3792391176672742840187796835728

Offset: 0

Alois P. Heinz, Aug 29 2017

nonn

			a(2) = 7: 1243, 1342, 1432, 2341, 2431, 3421, 4321.

Alois P. Heinz, Table of n, a(n) for n = 0..228

Cf. A186370.

Bisection (even part) of A303160.

b:= proc(n, k) option remember; `if`(k=0,
      `if`(n=0, 1, 0), `if`(k<0 or k>n, 0,
       k*b(n-1, k)+b(n-1, k-1)+(n-k+1)*b(n-1, k-2)))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..20);

b[n_, k_] := b[n, k] = If[k == 0, If[n == 0, 1, 0], If[k < 0 || k > n, 0, k*b[n - 1, k] + b[n - 1, k - 1] + (n - k + 1)*b[n - 1, k - 2]]];
a[n_] := b[2*n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 30 2019, after Alois P. Heinz *)

from sympy.core.cache import cacheit
@cacheit
def b(n, k): return (1 if n==0 else 0) if k==0 else 0 if k<0 or k>n else k*b(n - 1, k) + b(n - 1, k - 1) + (n - k + 1)*b(n - 1, k - 2)
def a(n): return b(2*n, n)
print([a(n) for n in range(31)]) # Indranil Ghosh, Aug 30 2017

a(n) = A186370(2n,n).

a(n) ~ c * d^n * n! * (n-1)!, where d = 3.4210546206711870249402157940795853513... and c = 0.32723781013647536133280275922604008889245... - Vaclav Kotesovec, Apr 29 2018

A303159 Number of permutations p of [2n+1] such that 0p has exactly n+1 alternating runs.

Original entry on oeis.org

1, 3, 43, 1344, 74211, 6384708, 789649750, 132789007200, 29145283614115, 8092186932120060, 2772830282722806978, 1149343084932146388144, 566844242187778610648334, 328043720353943611689811272, 220147053200818211779539712908, 169580070210721829547034445169024

Offset: 0

Alois P. Heinz, Apr 19 2018

nonn

			a(1) = 3: 132, 231, 321.

Alois P. Heinz, Table of n, a(n) for n = 0..227

Bisection (odd part) of A303160.

Cf. A186370.

b:= proc(n, k) option remember; `if`(k=0,
      `if`(n=0, 1, 0), `if`(k<0 or k>n, 0,
       k*b(n-1, k)+b(n-1, k-1)+(n-k+1)*b(n-1, k-2)))
    end:
a:= n-> b(2*n+1, n+1):
seq(a(n), n=0..20);

b[n_, k_] := b[n, k] = If[k == 0,
    If[n == 0, 1, 0], If[k < 0 || k > n, 0,
    k b[n-1, k] + b[n-1, k-1] + (n-k+1) b[n-1, k-2]]];
a[n_] := b[2n+1, n+1];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz *)

a(n) = A186370(2n+1,n+1).

a(n) ~ c * d^n * n!^2, where d = 3.4210546206711870249402157940795853513031388... and c = 0.974460718185930534652526741942010711752... - Vaclav Kotesovec, Apr 29 2018

A303160 Number of permutations p of [n] such that 0p has exactly ceiling(n/2) alternating runs.

Original entry on oeis.org

1, 1, 1, 3, 7, 43, 148, 1344, 6171, 74211, 425976, 6384708, 43979902, 789649750, 6346283560, 132789007200, 1219725741715, 29145283614115, 301190499710320, 8092186932120060, 92921064554444490, 2772830282722806978, 35025128774218944648, 1149343084932146388144

Offset: 0

Alois P. Heinz, Apr 19 2018

nonn

			a(2) = 1: 12.
a(3) = 3: 132, 231, 321.
a(4) = 7: 1243, 1342, 1432, 2341, 2431, 3421, 4321.

Alois P. Heinz, Table of n, a(n) for n = 0..456

Bisections give: A291677 (even part), A303159 (odd part).

Cf. A186370.

b:= proc(n, k) option remember; `if`(k=0,
      `if`(n=0, 1, 0), `if`(k<0 or k>n, 0,
       k*b(n-1, k)+b(n-1, k-1)+(n-k+1)*b(n-1, k-2)))
    end:
a:= n-> b(n, ceil(n/2)):
seq(a(n), n=0..25);

b[n_, k_] := b[n, k] = If[k == 0,
     If[n == 0, 1, 0], If[k < 0 || k > n, 0,
     k*b[n-1, k] + b[n-1, k-1] + (n-k+1)*b[n-1, k-2]]];
a[n_] := b[n, Ceiling[n/2]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 31 2021, after Alois P. Heinz *)

a(n) = A186370(n,ceiling(n/2)).

A241242 a(n) = -2^(2n+1)(E(2n+1, 1/2) + E(2n+1, 1) + 2(E(2n+2, 1/2) + E(2*n+2, 1))), where E(n,x) are the Euler polynomials.

Original entry on oeis.org

0, -3, 45, -1113, 42585, -2348973, 176992725, -17487754833, 2195014332465, -341282303124693, 64397376340013805, -14499110277050234553, 3840151029102915908745, -1182008039799685905580413, 418424709061213506712209285, -168805428822414120140493978273

Offset: 0

Peter Luschny, Apr 17 2014

sign

Cf. A000182, A000364, A028296, A240559, A147315, A186370.

A241242 := proc(n) e := n -> euler(n,1/2) + euler(n,1); -2^(2*n+1)*(e(2*n+1) + 2*e(2*n+2)) end: seq(A241242(n),n=0..15);

Array[-2^(2 # + 1)*(EulerE[2 # + 1, 1/2] + EulerE[2 # + 1, 1] + 2 (EulerE[2 # + 2, 1/2] + EulerE[2 # + 2, 1])) &, 16, 0] (* Michael De Vlieger, May 24 2018 *)

a(n) = A240559(2*n+1) = (-1)^n*A147315(2*n+1,2) = (-1)^n*A186370(2*n,2*n-1).
a(n) = Sum_{k=0..2*n+1} (-1)^(2*n+1-k)*binomial(2*n+1, k)*2^k*(E(k, 1/2) + 2*E(k+1, 0)) where E(n,x) are the Euler polynomials.
a(n) = Sum_{k=0..2*n+1} (-1)^(2*n+1-k)*binomial(2*n+1, k)*(skp(k, 0) + skp(k+1, -1)), where skp(n, x) are the Swiss-Knife polynomials A153641.
a(n) = Bernoulli(2*n + 2) * 4^(n+1) * (1 - 4^(n+1)) / (2*n + 2) - EulerE(2*n + 2), where EulerE(2*n) is A028296. - Daniel Suteu, May 22 2018
a(n) = (-1)^(n+1) * (A000182(n+1) - A000364(n+1)). - Daniel Suteu, Jun 23 2018

A186371 Number of up-down runs in all permutations of {1,2,...,n}.

Original entry on oeis.org

0, 1, 3, 13, 68, 420, 3000, 24360, 221760, 2237760, 24796800, 299376000, 3911846400, 55005350400, 828193766400, 13294689408000, 226663557120000, 4090405423104000, 77895546753024000, 1561112121913344000, 32844177110384640000, 723788347432550400000

Offset: 0

Emeric Deutsch and Ira M. Gessel, Mar 01 2011

The up-down runs of a permutation p are the alternating runs of the permutation p endowed with a 0 in the front. For example, 75814632 has 6 up-down runs: 07, 75, 58, 81, 146, and 632.

			a(3)=13 because the permutations 123, 132, 213, 231, 312, and 321 have a total of 1 + 2 + 3 + 2 + 3 + 2 = 13 up-down runs.

Cf. A186370, A097971.

[0,1] cat [Factorial(n)*(4*n+1)/6: n in [2..30]]; // Vincenzo Librandi, Sep 11 2015

0, 1, seq((1/6)*factorial(n)*(4*n+1), n = 2 .. 20);

Join[{0, 1}, Table[n! (4 n + 1)/6, {n, 2, 20}]] (* Vincenzo Librandi, Sep 11 2015 *)

a(n) = Sum_{k=1..n} k*A186370(n,k).
a(n) = n!*(4n+1)/6 for n>=2.
E.g.f.: g(z) = z(6-3z+z^2)/[6(1-z)^2].
D-finite with recurrence 4*a(n) +(-4*n-7)*a(n-1) +3*(n-1)*a(n-2)=0. - R. J. Mathar, Jul 22 2022

cp's OEIS Frontend

A240559 a(n) = -2^n*(E(n, 1/2) + E(n, 1) + (n mod 2)*2*(E(n+1, 1/2) + E(n+1, 1))), where E(n, x) are the Euler polynomials.

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A291677 Number of permutations p of [2n] such that 0p has exactly n alternating runs.

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A303159 Number of permutations p of [2n+1] such that 0p has exactly n+1 alternating runs.

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A303160 Number of permutations p of [n] such that 0p has exactly ceiling(n/2) alternating runs.

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A241242 a(n) = -2^(2*n+1)*(E(2*n+1, 1/2) + E(2*n+1, 1) + 2*(E(2*n+2, 1/2) + E(2*n+2, 1))), where E(n,x) are the Euler polynomials.

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A186371 Number of up-down runs in all permutations of {1,2,...,n}.

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A240559 a(n) = -2^n(E(n, 1/2) + E(n, 1) + (n mod 2)2*(E(n+1, 1/2) + E(n+1, 1))), where E(n, x) are the Euler polynomials.

A241242 a(n) = -2^(2n+1)(E(2n+1, 1/2) + E(2n+1, 1) + 2(E(2n+2, 1/2) + E(2*n+2, 1))), where E(n,x) are the Euler polynomials.