A209316 E.g.f.: Sum_{n>=0} a(n) * (cos(n^2*x) - sin(n^2*x)) * x^n/n! = 1/(1-x).
1, 1, 4, 57, 2456, 240205, 44616096, 14030856525, 6897867308800, 4999592004999705, 5107861266649227520, 7098997630368216900833, 13040338287878632604362752, 30913685990004537377333201253, 92695803952674372198927320920064, 345599063527286969179932122231749365
Offset: 0
Keywords
Examples
By definition, the coefficients a(n) satisfy: 1/(1-x) = 1 + 1*(cos(x)-sin(x))*x + 4*(cos(4*x)-sin(4*x))*x^2/2! + 57*(cos(9*x)-sin(9*x))*x^3/3! + 2456*(cos(16*x)-sin(16*x))*x^4/4! + 240205*(cos(25*x)-sin(25*x))*x^5/5! +...+ a(n)*(cos(n^2*x)-sin(n^2*x))*x^n/n! +...
Programs
-
PARI
a(n)=local(A=[1, 1], N); for(i=1, n, A=concat(A, 0); N=#A; A[N]=(N-1)!*(1-Vec(sum(m=0, N-1, A[m+1]*x^m/m!*(cos(m^2*x+x*O(x^N))-sin(m^2*x+x*O(x^N)))))[N])); A[n+1] for(n=0, 25, print1(a(n), ", "))
-
PARI
a(n)=if(n==0 || n==1, 1, n!+sum(k=1, n-1, (-1)^((n-k-1)\2)*a(k)*binomial(n, k)*k^(2*n-2*k))) for(n=0, 25, print1(a(n), ", "))
Formula
a(n) = n! + Sum_{k=1..n-1} (-1)^[(n-k-1)/2] * binomial(n,k) * k^(2*n-2*k) * a(k) for n>1 with a(0)=a(1)=1.