A219504 E.g.f.: Sum_{n>=0} a(n) * (cos(n*x)-sin(n*x)) * x^n/n! = 1/(1-x).

1, 1, 4, 33, 512, 13005, 494976, 26383917, 1876721664, 171728626617, 19650536857600, 2749029193911033, 461590186944847872, 91611982632843733125, 21215197576393952452608, 5669317752667727770720965, 1731566894935958076783067136, 599421136964093700021081229041

Offset

0,3

Comments

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

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

Links

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

Formula

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.

Example

E.g.f.: A(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 512*x^4/4! + 13005*x^5/5! +...
By definition, the coefficients a(n) satisfy:
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! +...

Mathematica

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 *)

Prog

(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]}
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^(n-k)))}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A218798.

Sequence in context: A101981 A002018 A368837 * A258180 A072754 A225609

Adjacent sequences: A219501 A219502 A219503 * A219505 A219506 A219507

Keyword

nonn

Author

Paul D. Hanna, Nov 21 2012

Status

approved