A219504 - OEIS

%I #15 Nov 18 2017 04:26:19

%S 1,1,4,33,512,13005,494976,26383917,1876721664,171728626617,

%T 19650536857600,2749029193911033,461590186944847872,

%U 91611982632843733125,21215197576393952452608,5669317752667727770720965,1731566894935958076783067136,599421136964093700021081229041

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

%C Compare to the LambertW identity: Sum_{n>=0} n^n * exp(-n*x) * x^n/n! = 1/(1-x).

%C Limit a(n)/A218798(n) = 2.30118311046652539351786883792086321360311554689487793288...

%H Paul D. Hanna, <a href="/A219504/b219504.txt">Table of n, a(n) for n = 0..100</a>

%F a(n) = n! + Sum_{k=1..n-1} (-1)^floor((n-k-1)/2) * binomial(n,k) * k^(n-k) * a(k) for n>1 with a(0)=a(1)=1.

%e E.g.f.: A(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 512*x^4/4! + 13005*x^5/5! +...

%e By definition, the coefficients a(n) satisfy:

%e 1/(1-x) = 1 + 1*(cos(x)-sin(x))*x + 4*(cos(2*x)-sin(2*x))*x^2/2! + 33*(cos(3*x)-sin(3*x))*x^3/3! + 512*(cos(4*x)-sin(4*x))*x^4/4! + 13005*(cos(5*x)-sin(5*x))*x^5/5! +...+ a(n)*(cos(n*x)-sin(n*x))*x^n/n! +...

%t a[0] := 1; a[1] := 1; a[n_] := n! + Sum[(-1)^(Floor[(n -k-1)/2]) *Binomial[n, k]*k^(n - k)*a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Nov 18 2017 *)

%o (PARI) {a(n)=local(A=[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*x+x*O(x^N))-sin(m*x+x*O(x^N)))))[N]));A[n+1]}

%o for(n=0,25,print1(a(n),", "))

%o (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^(n-k)))}

%o for(n=0, 25, print1(a(n), ", "))

%Y Cf. A218798.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 21 2012