A048990 -id:A048990 - cp's OEIS frontend

A343795 Number of Dumont permutations of the fourth kind of length 2n avoiding the pattern 312.

1, 1, 3, 10, 39, 174, 872, 4805, 28474, 178099, 1160173, 7803860, 53924841, 381640934, 2761331130, 20400560942, 153738854242, 1180631743440, 9229687049249, 73372263658451, 592476077260123, 4854377724124700, 40315729803287046, 339065862485375334, 2885324166565733641

Offset: 0

Alexander Burstein and Opel Jones, Apr 29 2021

Dumont permutations of the fourth kind are permutations of even length where all deficiencies (drops) are even values at even positions.

			For n=2, a(2)=3 counts the permutations 1234, 1342, 1432.

O. Jones, Enumeration of Dumont permutations avoiding certain four-letter patterns, Ph.D. thesis, Howard University, 2019.

A. Burstein and O. Jones, Enumeration of Dumont permutations avoiding certain four-letter patterns, arXiv:2002.12189 [math.CO], 2020.
A. Burstein, M. Josuat-Vergès, and W. Stromquist, New Dumont permutations, Pure Math. Appl. (Pu.M.A.) 21 (2010), no. 2, 177-206.

Cf. A000108 (permutations avoiding 312), A024492, A048990, A110501 (length 2n Dumont permutations of 4th kind).

seq(n)={my(h=sqrt(1-16*x + O(x*x^n)), F=sqrt((1-h)/(8*x)), G=(1-sqrt((1+h)/2))/(2*x), A=O(1)); forstep(k=n\3, 0, -1, my(f=Pol(F + O(x*x^k))); A = f/((1 - x*Pol(G + O(x^k)))^2 - x*f/(1 - x*Pol(G + O(x*x^k)) - x*f^2/(1 - x*A))) ); Vec(A + O(x*x^n))} \\ Andrew Howroyd, Apr 29 2021

Let F_k(x) be the truncation of the g.f. of A048990 to a polynomial of degree k. Let G_k(x) be the truncation of the g.f. of A024492 to a polynomial of degree k. Let G_{-1}(x) = 0. For k>=0, define A_k(x) recursively as follows: A_k(x) = F_k(x)/((1-x*G_{k-1}(x))^2-x*F_k(x)/(1-x*G_k(x)-x*F_k(x)^2/(1-x*A_{k+1}(x)))). Then A_0(x) is the g.f. of this sequence.

Terms a(12) and beyond from Andrew Howroyd, Apr 29 2021

A349648 Expansion of g.f.: Catalan(x)/Catalan(-x).

Original entry on oeis.org

1, 2, 2, 8, 14, 64, 132, 640, 1430, 7168, 16796, 86016, 208012, 1081344, 2674440, 14057472, 35357670, 187432960, 477638700, 2549088256, 6564120420, 35223764992, 91482563640, 493132709888, 1289904147324, 6979724509184, 18367353072152, 99710350131200

Offset: 0

Alexander Burstein, Nov 23 2021

Cf. A000108, A001622, A048990 (bijection), A052707 (bijection), A006318, A079489, A246062, A333564.

gf:= (c-> c(x)/c(-x))(x-> hypergeom([1/2, 1], [2], 4*x)):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..35);  # Alois P. Heinz, Nov 23 2021

CoefficientList[Series[(1-Sqrt[1-4x])/(Sqrt[1+4x]-1),{x,0,24}],x]

a(2*n) = A048990(n) = A000108(2*n), n>=0.
a(2*n+1) = A052707(n+1) = 2^(2*n+1)*A000108(n), n>=0.
G.f.: A(x) = C(x)/C(-x) = (1 - sqrt(1 - 4*x))/(sqrt(1 + 4*x) - 1), where C(x) is the g.f. of A000108.
G.f.: A(x) = F(x^2) + 2*x*F(x^2)^2 = (C(x) + C(-x))/2 + 2*x*C(4*x^2), where F(x) is the g.f. of A048990.
G.f.: A(-x) = 1/A(x).
G.f.: A(x) = R(x*C(-x)^2) = 1/R(-x*C(x)^2), where R(x) is the g.f. of A006318.
G.f.: A(x) = (1 + x*C(x)*C(-x))/(1 - x*C(x)*C(-x)), see A079489 for the expansion of C(x)*C(-x).
D-finite with recurrence n*(n-1)*(n+1)*a(n) -4*(n-1)*(8*n^2-32*n+35)*a(n-2) +64*(2*n-5)*(2*n-7)*(n-4)*a(n-4)=0. - R. J. Mathar, Mar 06 2022
Sum_{n>=0} 1/a(n) = 28/15 + 2*Pi/(9*sqrt(3)) + 64*arcsin(1/4)/(75*sqrt(15)) - 12*log(phi)/(25*sqrt(5)), where phi is the golden ratio (A001622). - Amiram Eldar, Apr 20 2023
G.f.: A(x) = exp( Sum_{n >= 1} binomial(4*n-2,2*n-1)*x^(2*n-1)/(2*n-1) ). - Peter Bala, Apr 28 2023

A376075 Number of North-East lattice paths from (0,0) to (n,n) that do not cross the diagonal y = x at any even point (2k,2k).

Original entry on oeis.org

1, 2, 6, 14, 52, 140, 558, 1598, 6604, 19588, 82780, 251212, 1077992, 3324760, 14427422, 45039422, 197122524, 621205076, 2737289748, 8691699524, 38510822360, 123045322024, 547682980716, 1759017606220, 7859796084984, 25355507376808, 113670929821304

Offset: 0

John Tyler Rascoe, Oct 08 2024

			The path NENNEENE does not cross y = x, so it is counted under a(4) = 52.
The path NENNENNEEEEN crosses y = x at points (1,1) and (5,5), so it is counted under a(6) = 558.

Cf. A000108, A000984, A024492, A048990, A268400, A268429.

C(x) = {(1-sqrt(1-4*x))/(2*x)}
A(x) ={C(4*x)*C((x)*C(4*x))}
B(x) = {sqrt(C(4*x))}
D(x) = {1/sqrt(1-4*x)}
E_x(N) = {my(x='x+O('x^N));  Vec(D(x)-2*((C(x)-1)*((x*A(x^2))^2-B(x^2)^2+3*B(x^2)-2))/((2-B(x^2))*(2-C(x))))}
E_x(30)

G.f. D(x) - 2*((C(x) - 1)*((x*A(x^2))^2 - B(x^2)^2 + 3*B(x^2) - 2))/((2 - B(x^2))*(2 - C(x))), where A(x), B(x), C(x), and D(x) are the g.f.s for A024492, A048990, A000108, and A000984.

A382403 a(n) = Sum_{k=0..n} A039599(n,k)^3.

Original entry on oeis.org

1, 2, 36, 980, 33040, 1268568, 53105976, 2364239592, 110206067400, 5323547715200, 264576141331216, 13458185494436592, 697931136204820336, 36789784967375728400, 1966572261077797609200, 106400946932857148590800, 5817987630644593688220600, 321105713814359742307398480

Offset: 0

Seiichi Manyama, Mar 24 2025

nonn

Let b_k(n) = Sum_{j=0..n} A039599(n,j)^k. b_1(n) = binomial(2*n,n) = A000984(n) and b_2(n) = binomial(4*n,2*n)/(2*n+1) = A048990(n).

Pedro J. Miana, Hideyuki Ohtsuka, and Natalia Romero, Sums of powers of Catalan triangle numbers, arXiv:1602.04347 [math.NT], 2016.

Cf. A000984, A048990.

Cf. A039599, A112029.

a039599(n, k) = (2*k+1)/(n+k+1)*binomial(2*n, n-k);
a(n) = sum(k=0, n, a039599(n, k)^3);

a(n) = binomial(2*n,n) * (4 * binomial(2*n,n)^2 - 3 * A112029(n)).

cp's OEIS Frontend

A343795 Number of Dumont permutations of the fourth kind of length 2n avoiding the pattern 312.

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A349648 Expansion of g.f.: Catalan(x)/Catalan(-x).

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A376075 Number of North-East lattice paths from (0,0) to (n,n) that do not cross the diagonal y = x at any even point (2*k,2*k).

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A382403 a(n) = Sum_{k=0..n} A039599(n,k)^3.

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A376075 Number of North-East lattice paths from (0,0) to (n,n) that do not cross the diagonal y = x at any even point (2k,2k).