A186364 -id:A186364 - cp's OEIS frontend

A348580 Expansion of e.g.f. exp(x) / (1 - sin(x)).

1, 2, 5, 15, 53, 217, 1015, 5355, 31513, 204857, 1458875, 11299695, 94600373, 851419597, 8198959735, 84124450035, 916270051633, 10559066809937, 128362804540595, 1641730799916375, 22037407161945293, 309782122281453877, 4551072446448773455, 69747642031977698715

Offset: 0

Ilya Gutkovskiy, Oct 24 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..484

Cf. A000111, A000667, A186364, A330046, A348587.

b:= proc(u, o) option remember; `if`(u+o=0, 1,
      add(b(o-1+j, u-j), j=1..u))
    end:
a:= n-> add(binomial(n, k)*b(k+1, 0), k=0..n):
seq(a(n), n=0..23);  # Alois P. Heinz, Oct 24 2021

nmax = 23; CoefficientList[Series[Exp[x]/(1 - Sin[x]), {x, 0, nmax}], x] Range[0, nmax]!

my(x='x+O('x^40)); Vec(serlaplace(exp(x)/(1-sin(x)))) \\ Michel Marcus, Oct 24 2021

a(n) = Sum_{k=0..n} binomial(n,k) * A000111(k+1).

a(n) ~ 2^(n + 7/2) * n^(n + 3/2) / (Pi^(n + 3/2) * exp(n - Pi/2)). - Vaclav Kotesovec, Oct 25 2021

A186363 Triangle read by rows: T(n,k) is the number of cycle-up-down permutations of {1,2,...,n} having k fixed points (0 <= k <= n). A permutation is said to be cycle-up-down if it is a product of up-down cycles. A cycle (b(1), b(2), ...) is said to be up-down 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, 3, 0, 1, 5, 4, 6, 0, 1, 15, 25, 10, 10, 0, 1, 71, 90, 75, 20, 15, 0, 1, 341, 497, 315, 175, 35, 21, 0, 1, 1945, 2728, 1988, 840, 350, 56, 28, 0, 1, 12135, 17505, 12276, 5964, 1890, 630, 84, 36, 0, 1, 84091, 121350, 87525, 40920, 14910, 3780, 1050, 120, 45, 0, 1

Offset: 0

Emeric Deutsch, Feb 28 2011

Sum of entries in row n is A000111(n+1) (the Euler or up-down numbers).
T(n,0) = A186364(n).
T(n,k) = T(n-k,0)*binomial(n,k).
Sum_{k=0..n} k*T(n,k) = A186365(n).

			T(3,1)=3 because we have (1)(23), (12)(3), and (13)(2).
T(4,2)=6 because we have (1)(2)(34), (1)(23)(4), (1)(24)(3), (12)(3)(4), (13)(2)(4), and (14)(2)(3).
Triangle starts:
   1;
   0,  1;
   1,  0,  1;
   1,  3,  0,  1;
   5,  4,  6,  0,  1;
  15, 25, 10, 10,  0,  1;

E. Deutsch and S. Elizalde, Cycle up-down permutations, arXiv:0909.5199v1 [math.CO].

Cf. A186364, A186365.

G := exp((x-1)*z)/(1-sin(z)): Gser := simplify(series(G, z = 0, 16)): 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], x, j), j = 0 .. n) end do; # yields sequence in triangular form

T[n_, k_] := T[n, k] = If[k == 0, SeriesCoefficient[Exp[-x]/(1 - Sin[x]), {x, 0, n}] n!, T[n - k, 0] Binomial[n, k]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 24 2018 *)

E.g.f. = exp((x-1)z)/(1-sin z).

The trivariate e.g.f. H(t,s,z) of the cycle-up-down permutations of {1,2,...,n} with respect to size (marked by z), number of cycles (marked by t), and number of fixed points (marked by x) is given by H(t,x,z) = exp((x-1)*t*z)/(1-sin(z))^t.

A321632 Expansion of e.g.f. (1 + sin(x))/exp(x).

Original entry on oeis.org

1, 0, -1, 1, 1, -5, 9, -9, 1, 15, -31, 31, 1, -65, 129, -129, 1, 255, -511, 511, 1, -1025, 2049, -2049, 1, 4095, -8191, 8191, 1, -16385, 32769, -32769, 1, 65535, -131071, 131071, 1, -262145, 524289, -524289, 1, 1048575, -2097151, 2097151, 1, -4194305, 8388609, -8388609

Offset: 0

Paolo P. Lava, Nov 16 2018

A140323(n) = |a(4*n-1)| = |a(4*n-2)|, A247281(n) = |a(4*n+1)|.

The absolute values of the coefficients of the expansion of the reciprocal of this function are listed in A186364.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-3,-4,-2).

Cf. A140323, A186364, A247281.

seq(factorial(n)*coeff(series((1+sin(x))/exp(x),x=0,48),x,n),n=0..47);

With[{nn=50},CoefficientList[Series[(1+Sin[x])/Exp[x],{x,0,nn}],x] Range[ 0,nn]!] (* or *) LinearRecurrence[{-3,-4,-2},{1,0,-1},50] (* Harvey P. Dale, Jul 21 2021 *)

Vec((1 + 3*x + 3*x^2) / ((1 + x)*(1 + 2*x + 2*x^2)) + O(x^40)) \\ Colin Barker, Nov 16 2018

a(4*k)   = 1;
a(4*k+1) = (-4)^k - 1;
a(4*k+2) = -2*a(4*k+1) - 1 = -2*(-4)^k + 1;
a(4*k+3) =  2*a(4*k+1) + 1 =  2*(-4)^k - 1.
From Colin Barker, Nov 16 2018: (Start)
G.f.: (1 + 3*x + 3*x^2) / ((1 + x)*(1 + 2*x + 2*x^2)).
a(n) = (-1)^n + i/2*((-1-i)^n - (-1+i)^n), where i=sqrt(-1).
a(n) = -3*a(n-1) - 4*a(n-2) - 2*a(n-3) for n>2. (End)

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A348580 Expansion of e.g.f. exp(x) / (1 - sin(x)).

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A321632 Expansion of e.g.f. (1 + sin(x))/exp(x).

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