A180473 Expansion of o.g.f. x*s(x)/(1-x*s(x)-x^2*s(x)^2), where s(x) is the o.g.f. of the little Schroeder numbers (A001003).

1, 2, 7, 27, 114, 509, 2365, 11318, 55411, 276231, 1397430, 7156089, 37023225, 193229466, 1016141199, 5378940051, 28638955098, 153267403397, 824014568581, 4448456379134, 24104579252971, 131055735586767, 714741620026542, 3908997981612017, 21434123083817329

Offset: 1

Author

Vladimir Kruchinin, Sep 07 2010

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..500
• Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Programs

Maxima

a(n):=sum(k/(2^k*n)*sum(binomial(n,j)*2^(n-j)*(-1)^j*binomial(2*n-k-j-1,n-1),j,0,n-k)*fib(k),k,1,n);

PARI

seq(n)={my(p=x*(1+x-sqrt(1 - 6*x + x^2 + O(x*x^n)))/(4*x)); Vec(p/(1 - p - p^2))} \\ Andrew Howroyd, Apr 17 2021

Formula

a(n) = Sum_{k=1..n} (k/(2^k*n))*(Sum_{j=0..n-k} binomial(n,j)*2^(n-j)*(-1)^j*binomial(2*n-k-j-1, n-1))*Fibonacci(k).

Extensions

Terms a(21) and beyond from Andrew Howroyd, Apr 17 2021