A218798 E.g.f.: Sum_{n>=0} a(n) * (cos(n*x)-sin(n*x)) * x^n/n! = 1 + x.

1, 1, 2, 15, 224, 5665, 215136, 11466903, 815542272, 74626924833, 8539305740800, 1194616544819423, 200588161750990848, 39810821495648979009, 9219256372209251966976, 2463653490073311266825895, 752468103154551980520636416, 260483894045203867466646001345

Offset: 0

Author

Paul D. Hanna, Nov 21 2012

Keywords

• nonn

Comments

Compare to the LambertW identity:

1 + Sum_{n>=1} n^(n-1) * exp(-n*x) * x^n/n! = 1 + x.

Limit A219504(n)/A218798(n) = 2.30118311046652539351786883792086321360311554689487793288...

Examples

			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! +...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A219504.

Programs

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),", "))

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.