OFFSET
0,3
COMMENTS
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..100
FORMULA
a(n) = Sum_{k=1..n-1} (-1)^[(n-k-1)/2] * binomial(n,k) * k^(n-k) * a(k) for n>1 with a(0)=a(1)=1.
EXAMPLE
E.g.f.: A(x) = 1 + x + 2*x^2/2! + 15*x^3/3! + 224*x^4/4! + 5665*x^5/5! +...
By definition, the coefficients a(n) satisfy:
1+x = 1 + 1*(cos(x)-sin(x))*x + 2*(cos(2*x)-sin(2*x))*x^2/2! + 15*(cos(3*x)-sin(3*x))*x^3/3! + 224*(cos(4*x)-sin(4*x))*x^4/4! + 5665*(cos(5*x)-sin(5*x))*x^5/5! +...+ a(n)*(cos(n*x)-sin(n*x))*x^n/n! +...
PROG
(PARI) a(n)=local(A=[1, 1], N); for(i=1, n, A=concat(A, 0); N=#A; A[N]=(N-1)!*(-Vec(sum(m=0, N-1, A[m+1]*x^m/m!*(cos(m*x+x*O(x^N))-sin(m*x+x*O(x^N)))))[N])); A[n+1]
for(n=0, 25, print1(a(n), ", "))
(PARI) a(n)=if(n<2, 1, sum(k=1, n-1, (-1)^((n-k-1)\2)*a(k)*binomial(n, k)*k^(n-k)))
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 21 2012
STATUS
approved