A001586 -id:A001586 - cp's OEIS frontend

A385421 Expansion of e.g.f. 1/(1 - arcsin(2*x))^(1/2).

1, 1, 3, 19, 153, 1689, 21867, 343995, 6114993, 124933425, 2820098643, 70897706595, 1939085791305, 57898697121225, 1859540697970875, 64312039377723915, 2371651908598754145, 93246340110716523105, 3882169166979871734435, 171024539858087082582195

Offset: 0

Seiichi Manyama, Jun 28 2025

nonn

Cf. A189780, A385422.

Cf. A001147, A001586, A385343.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-asin(2*x))^(1/2)))

a(n) = Sum_{k=0..n} A001147(k) * 2^(n-k) * A385343(n,k).

a(n) ~ sqrt(sin(2)) * 2^n * n^n / (exp(n) * sin(1)^(n+1)). - Vaclav Kotesovec, Jun 28 2025

A201990 E.g.f. satisfies: A(x) = 1/(cos(xA(x)^2) - sin(xA(x)^2)).

Original entry on oeis.org

1, 1, 7, 95, 1969, 55201, 1956375, 83935039, 4230528353, 245059707841, 16043680004903, 1171567218325151, 94415150206330641, 8323801562833775201, 796927800013656980791, 82342529545666235490431, 9132868398860301753027265, 1082287792241161814647419265

Offset: 0

Paul D. Hanna, Dec 07 2011

nonn

Compare e.g.f. to: Sum_{n>=0} (2*n+1)^(n-1)*x^n/n! = sqrt((1/x)*Series_Reversion(x*(cosh(x) - sinh(x))^2)).
The radius of convergence r of e.g.f. A(x) is given by:
r = t*(cos(t) - sin(t))^2 where t = (1 - sin(2*t))/(2*cos(2*t)), so that:
r = 0.13127 35638 55724 99317 13322 82818 86189 50670 52604 32023 ...
t = 0.27798 42153 59698 32056 15352 87789 00442 74782 64480 84947 ...
Further, A(r) = 1/(cos(t) - sin(t)), thus
A(r) = 1.45519 57921 91350 02891 97122 64456 17664 48847 98244 19461 ...

			E.g.f.: A(x) = 1 + x + 7*x^2/2! + 95*x^3/3! + 1969*x^4/4! + 55201*x^5/5! +...
where
1/(cos(x)-sin(x)) = 1 + x + 3*x^2/2! + 11*x^3/3! + 57*x^4/4! + 361*x^5/5! + 2763*x^6/6! + 24611*x^7/7! +...+ A001586(n)*x^n/n! +...
The coefficients of x^n/n! in odd powers of G(x) = 1/(cos(x)-sin(x)) begin:
G^1: [(1), 1, 3, 11, 57, 361, 2763, 24611, ..., A001586(n), ...];
G^3: [1,(3), 15, 93, 705, 6243, 63375, 724413, ...];
G^5: [1, 5,(35), 295, 2905, 32525, 407435, 5638495, ...];
G^7: [1, 7, 63,(665), 8001, 107527, 1592703, 25738265, ...];
G^9: [1, 9, 99, 1251, (17721), 276849, 4716459, 86873211, ...];
G^11:[1, 11, 143, 2101, 34177, (607211), 11668943, 240764821, ...];
G^13:[1, 13, 195, 3263, 59865, 1190293,(25432875), 580193783, ...];
G^15:[1, 15, 255, 4785, 97665, 2146575, 50429055,(1259025585), ...]; ...
where coefficients in parenthesis form the initial terms of this sequence:
[1/1, 3/3, 35/5, 665/7, 17721/9, 607211/11, 25432875/13, 1259025585/15, ...].

Cf. A201923, A001586.

CoefficientList[Sqrt[1/x*InverseSeries[Series[x*(Cos[x] - Sin[x])^2, {x, 0, 21}], x]],x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 12 2014 *)

{a(n)=local(X=x+x*O(x^n));n!*polcoeff(sqrt(1/x*serreverse(x*(cos(X)-sin(X))^2)),n)}

{a(n)=local(A=1+x,X=x+x*O(x^n));for(i=1,n,A=1/(cos(X*A^2) - sin(X*A^2)));n!*polcoeff(A,n)}

{a(n)=local(X=x+x*O(x^n),A001586=1/(cos(X)-sin(X)));n!*polcoeff(A001586^(2*n+1),n)/(2*n+1)}

E.g.f. satisfies: A( x*(cos(x) - sin(x))^2 ) = 1/(cos(x) - sin(x)).
E.g.f: sqrt( (1/x) * Series_Reversion( x*(cos(x) - sin(x))^2 ) ).
a(n) = [x^n/n!] 1/(cos(x)-sin(x))^(2*n+1) / (2*n+1).
a(n) ~ sqrt((t*cos(2*t))/(2*cos(2*t)+4*t*(3+sin(2*t)))) * n^(n-1) / (exp(n) * r^(n+1/2)), where r and t were described above. - Vaclav Kotesovec, Jan 12 2014

A335335 Irregular triangle T(n,k) of Arnold numbers with n>=1 and 1<= abs(k) <= n.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 0, 2, 3, 3, 4, 4, 0, 4, 8, 11, 11, 14, 16, 16, 0, 16, 32, 46, 57, 57, 68, 76, 80, 80, 0, 80, 160, 236, 304, 361, 361, 418, 464, 496, 512, 512, 0, 512, 1024, 1520, 1984, 2402, 2763, 2763, 3124, 3428, 3664, 3824, 3904, 3904, 0, 3904, 7808, 11632, 15296, 18724, 21848, 24611, 24611, 27374, 29776, 31760, 33280, 34304, 34816, 34816

Offset: 1

Michel Marcus, Jun 02 2020

			Triangle begins:
                        1,   1,
                   0,   1,   1,   2,
              0,   2,   3,   3,   4,   4,
         0,   4,   8,  11,  11,  14,  16,  16,
    0,  16,  32,  46,  57,  57,  68,  76,  80,  80,
0, 80, 160, 236, 304, 361, 361, 418, 464, 496, 512, 512,

Heesung Shin and Jiang Zeng, More bijections for Entringer and Arnold families, arXiv:2006.00507 [math.CO], 2020.

Cf. A185356, A202690, A202815, A202816.

Cf. A001586 (row sums).

T(n, k) = {if ((n==1) && (k==1), return (1)); if ((n+k) == 0, if (n==1, return(1), return (0))); if ((n>=k) && (k>1), return(T(n, k-1) + T(n-1, 1-k))); if ((k==1) && (n>k), return(T(n,-1))); if ((-1>=k) && (k>=-n), return(T(n, k-1) + T(n-1, -k)));}
tabf(nn) = {for (n=1, nn, for (k=-n, -1, print1(T(n,k), ", ");); for (k=1, n, print1(T(n,k), ", ");); print;);}

T(n,k) is defined by T(1,1) = T(1,-1) = 1, T(n,-n) = 0 (n >= 2), and the recurrence
T(n,k) = T(n,k-1) + T(n-1,-k+1) if n >= k > 1,
T(n,k) = T(n,-1) if n > k = 1,
T(n,k) = T(n,k-1) + T(n-1,-k) if -1 >= k > -n.

A347930 3-Springer numbers.

Original entry on oeis.org

1, 1, 3, 16, 88, 625, 5527, 55760, 640540, 8329326, 120212331, 1905939913, 32987637967, 618591571085, 12489644875037, 270193806214360, 6235154917414954, 152875655211527878, 3968729594485785289, 108754865309750398187, 3137052120203959610759

Offset: 2

Alejandro H. Morales, Sep 19 2021

nonn

a(n) is also the volume of a certain flow polytope.

Arvind Ayyer, Matthieu Josuat-Vergès, and Sanjay Ramassamy, Extensions of partial cyclic orders and consecutive coordinate polytopes, Ann. H. Lebesgue, 3 (2020), 275-297.
R. S. Gonzalez D'Leon, A. H. Morales, C. R. H. Hanusa, and M. Yip, Column convex matrices, G-cyclic orders, and flow polytopes, arXiv:2107.07326 [math.CO], 2021.
S. Ramassamy, Extensions of partial cyclic orders, Euler numbers and multidimensional boustrophedons, Electron. J. Combin., 25 (2018), #P1.66.

Cf. A001586, A008282, A000111.

wcomps:=proc(n,k)
       option remember;
local ocomps,ncomps,i;
ocomps:=combinat:-composition(n+k,k);
ncomps:={};
for i from 1 to nops(ocomps) do
   ncomps:=ncomps union{[seq(ocomps[i][j]-1,j=1..k)]};
end do;
return [op(ncomps)];
end proc:
b:=proc(s) option remember;
   local k;
   k := nops(s);
   if s = [seq(0,i=1..k)] then
      return(1);
   elif s[1]>0 then
      return(add(b([s[2]+j,op(s[3..k]),s[1]-j-1]),j=0..s[1]-1));
   else
      return(0);
   end if;
end proc:a:=proc(n)   local N,S:   N := n-2;   S := wcomps(N,3);   return add(combinat:-multinomial(N,op(s))*b(s), s in S);end proc:seq(a(n),n=2..10);

a(n) = Sum_{(x,y,z), x+y+z=n-2} ((n-2)!/(x!*y!*z!))*b(x,y,z), where b(x,y,z) are the 3-Entringer numbers defined by Ramassamy.

A385895 Table read by rows: T(n, k) = T(n, k-1) + m * T(n-1, n-k) for k > 1, T(n, 1) = T(n-1, n-1), and T(n, 0) = 0^n, for m = 2.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 3, 0, 3, 9, 11, 11, 0, 11, 33, 51, 57, 57, 0, 57, 171, 273, 339, 361, 361, 0, 361, 1083, 1761, 2307, 2649, 2763, 2763, 0, 2763, 8289, 13587, 18201, 21723, 23889, 24611, 24611, 0, 24611, 73833, 121611, 165057, 201459, 228633, 245211, 250737, 250737

Offset: 0

Peter Luschny, Jul 20 2025

The sequence extends the generalized Euler numbers A001586 to a regular table by a parametrized Seidel transformation (see the Python program) that for the case m = 1 leads to the Euler-Bernoulli numbers A008281.

			Triangle begins:
  [0] 1;
  [1] 0,    1;
  [2] 0,    1,    1;
  [3] 0,    1,    3,     3;
  [4] 0,    3,    9,    11,    11;
  [5] 0,   11,   33,    51,    57,    57;
  [6] 0,   57,  171,   273,   339,   361,   361;
  [7] 0,  361, 1083,  1761,  2307,  2649,  2763,  2763;
  [8] 0, 2763, 8289, 13587, 18201, 21723, 23889, 24611, 24611;

Cf. A001586 (main diagonal), A123110 (m=0), A008281 (m=1), this sequence (m=2).

T := proc(n, k) option remember; ifelse(k = 0, 0^n, ifelse(k = 1, T(n-1, n-1), T(n, k-1) + 2*T(n-1, n-k))) end: seq(seq(T(n, k), k = 0..n), n = 0..9);

T[n_, k_] := T[n, k] =
  Which[
    k == 0, Boole[n == 0],
    k == 1, T[n - 1, n - 1],
    True, T[n, k - 1] + 2*T[n - 1, n - k]
  ];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

from functools import cache
@cache
def seidel(n: int, m: int) -> list[int]:
    if n == 0: return [1]
    rowA = seidel(n - 1, m)
    row = [0] + rowA
    row[1] = row[n]
    for k in range(2, n + 1):
        row[k] = row[k - 1] + m * rowA[n - k]
    return row
def A385895row(n: int) -> list[int]: return seidel(n, 2)
for n in range(9): print(A385895row(n))

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A385421 Expansion of e.g.f. 1/(1 - arcsin(2*x))^(1/2).

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A201990 E.g.f. satisfies: A(x) = 1/(cos(x*A(x)^2) - sin(x*A(x)^2)).

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A335335 Irregular triangle T(n,k) of Arnold numbers with n>=1 and 1<= abs(k) <= n.

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A347930 3-Springer numbers.

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A385895 Table read by rows: T(n, k) = T(n, k-1) + m * T(n-1, n-k) for k > 1, T(n, 1) = T(n-1, n-1), and T(n, 0) = 0^n, for m = 2.

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A201990 E.g.f. satisfies: A(x) = 1/(cos(xA(x)^2) - sin(xA(x)^2)).