OFFSET
0,3
COMMENTS
Equals base sequence of pendular trinomial triangle A119369; iterated convolutions of this sequence with the central terms (A119371) generates all diagonals of A119369. For example: A119372 = A119370 * A119371; A119373 = A119370^2 * A119371.
Diagonal sums of number array A133336. - Philippe Deléham, Nov 09 2009
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Stoyan Dimitrov, On permutation patterns with constrained gap sizes, arXiv:2002.12322 [math.CO], 2020.
Hannah Golab, Pattern avoidance in Cayley permutations, Master's Thesis, Northern Arizona Univ. (2024). See p. 41.
FORMULA
G.f.: A(x) = ((1+x^2) - sqrt( (1+x^2)^2 - 4*x*(1+x) ))/(2*x*(1+x)). Equals the inverse binomial transform of A104547.
Recurrence: (n+1)*a(n) = 3*(n-1)*a(n-1) + 6*(n-1)*a(n-2) + 2*(n-2)*a(n-3) - (n-5)*a(n-4) - (n-5)*a(n-5). - Vaclav Kotesovec, Sep 11 2013
a(n) ~ sqrt(-z^2-3*z+1)*(4+2*z-z^3)^(n+1)*(-z^3+z^2+z+3) / (8*sqrt(Pi) * n^(3/2)), where z = 1/(2*sqrt(3/(4+(280-24*sqrt(129))^(1/3) + 2*(35 + 3*sqrt(129))^(1/3)))) - 1/2*sqrt(8/3-1/3*(280-24*sqrt(129))^(1/3) - 2/3*(35+3*sqrt(129))^(1/3) + 8*sqrt(3/(4+(280-24*sqrt(129))^(1/3) + 2*(35 + 3*sqrt(129))^(1/3)))) = 0.225270426... is the root of the equation 1-2*z^2+z^4-4*z=0. - Vaclav Kotesovec, Sep 11 2013
G.f.: 1/G(0) where G(k) = 1 - q/(1 - (q + q^2) / G(k+1) ). - Joerg Arndt, Dec 06 2014
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k-1,k) * binomial(2*n-3*k+1,n-2*k)/(2*n-3*k+1). - Seiichi Manyama, Aug 28 2023
Conjecture: A(x) = 1 + x*exp(Sum_{n >= 1} g(n, x)*x^n/n), where g(n, x) = Sum_{k = 0..n} binomial(n, k)^2*(1 + x)^k. Cf. A105633 and A167638. - Peter Bala, Sep 10 2024
EXAMPLE
A(x) = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 64*x^5 + 225*x^6 + 816*x^7 +...
x*A(x)^2 = x + 2*x^2 + 5*x^3 + 16*x^4 + 54*x^5 + 190*x^6 + 690*x^7 +...
x^2*( A(x)^2 - A(x) ) = 1*x^3 + 3*x^4 + 10*x^5 + 35*x^6 + 126*x^7 +...
MAPLE
m:= 30;
S:= series( (1+x^2 -sqrt(1-4*x-2*x^2+x^4))/(2*x*(1+x)), x, m+1);
seq(coeff(S, x, j), j = 0..m); # G. C. Greubel, Mar 17 2021
MATHEMATICA
CoefficientList[Series[((1+x^2)-Sqrt[(1+x^2)^2-4*x*(1+x)])/(2*x*(1+x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 11 2013 *)
PROG
(PARI) {a(n)=polcoeff(2/((1+x^2)+sqrt((1+x^2)^2-4*x*(1+x)+x*O(x^n))), n)}
(Sage)
def A119370_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( (1+x^2 -sqrt(1-4*x-2*x^2+x^4))/(2*x*(1+x)) ).list()
A119370_list(30) # G. C. Greubel, Mar 17 2021
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (1+x^2 -Sqrt(1-4*x-2*x^2+x^4))/(2*x*(1+x)) )); // G. C. Greubel, Mar 17 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul D. Hanna, May 16 2006
STATUS
approved