A136576 -id:A136576 - cp's OEIS frontend

A136577 Conjectured Hankel transform of A136576(n+1).

1, 1, 0, -16, -256, -4096, 0, 16777216, 4294967296, 1099511627776, 0, -1152921504606846976, -4722366482869645213696, -19342813113834066795298816, 0, 5192296858534827628530496329220096, 340282366920938463463374607431768211456

Offset: 0

Paul Barry, Jan 08 2008

a(n+1) is the Hankel transform of A086616. [From Paul Barry, Jun 03 2009]

a(n)=4^floor(n^2/2)*((1/2-sqrt(2)/2)*cos(3*pi*n/4)+(1/2+sqrt(2)/2)*cos(pi*n/4));

A069731 Number of unicursal planar maps with n edges rooted at a vertex of odd valency (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

Original entry on oeis.org

1, 5, 28, 168, 1056, 6864, 45760, 311168, 2149888, 15049216, 106502144, 760729600, 5477253120, 39710085120, 289650032640, 2124100239360, 15651264921600, 115819360419840, 860372391690240

Offset: 1

Valery A. Liskovets, Apr 07 2002

V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Cf. A069724, A069720, A000108, A003645.

Cf. A000346, A001006, A136576.

Z:=-(1-4*z-sqrt(1-4*z))/sqrt(1-4*z)/64: Zser:=series(Z, z=0, 32): seq(coeff(Zser*2^(n+1), z, n), n=3..24); # Zerinvary Lajos, Jan 01 2007

Table[2^(n-2) CatalanNumber[n+1], {n, 1, 19}] (* Jean-François Alcover, Aug 28 2019 *)

a(n) = 2^(n-2)*C_(n+1), where C_n stands for the Catalan numbers (A000108).
a(n) = A003645(n+2)/4.
D-finite with recurrence: 4*(2*n+1)*a(n-1) - (n+2)*a(n) = 0, a(1) = 1. - Georg Fischer, May 23 2021
From Peter Bala, Apr 29 2024: (Start)
a(n) = Sum_{k = 0..n} binomial(n, 2*k)*Catalan(k)*4^(n-k-1).
O.g.f.: A(x) = (1 - 4*x - 8*x^2 - sqrt(1 - 8*x))/(32*x^2).
A(x) = series reversion of x*c(-x)/(1 + 4*x), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108 and c(-x)/(1 + 4*x) is the g.f. of (-1)^n*A000346(n). (End)

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A136577 Conjectured Hankel transform of A136576(n+1).

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A069731 Number of unicursal planar maps with n edges rooted at a vertex of odd valency (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

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