A100065 E.g.f. A(x) satisfies: Sum_{k=0..n} (A(x)^n)_k/k! = [exp(n)] (integer floor of e^n) for all n>=0, where (A(x)^n)_k/k! is the coefficient of x^k in A(x)^n.

1, 1, 3, -3, -57, 369, 3861, -76617, -413775, 26879103, -52349229, -13272657111, 163271276343, 8454393401121, -248550519324411, -5938622542044441, 389600359347114081, 2878180608333977343, -677734293694386004317, 5701654525891937630409

Offset

0,3

Comments

See triangle A100064 of initial coefficients of successive powers of the e.g.f. for this sequence.

Links

Table of n, a(n) for n=0..19.

Example

List the coefficients of powers of e.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n!:
A(x)^0: [1,__0,0,0,0,0,0,0,0,...],
A(x)^1: [1,1,__3,-3,-57,369,3861,-76617,-413775,...],
A(x)^2: [1,2,8,__12,-84,-12,7200,-40716,-1301328,...],
A(x)^3: [1,3,15,51,__27,-513,4077,33237,-1211895,...],
A(x)^4: [1,4,24,120,408,__216,-3168,45576,-202176,...],
A(x)^5: [1,5,35,225,1215,4365,__1485,-27765,440865,...],...
then for each row n, Sum_{k=0..n} (A(x)^n)_k/k! = [exp(n)]:
[exp(0)] = 1 = 1
[exp(1)] = 1+1 = 2
[exp(2)] = 1+2+8/2! = 7
[exp(3)] = 1+3+15/2!+51/3! = 20
[exp(4)] = 1+4+24/2!+120/3!+408/4! = 54
[exp(5)] = 1+5+35/2!+225/3!+1215/4!+4365/5! = 148

Prog

(PARI) {a(n)=local(A, C, F, G); if(n==0, A=1, F=sum(k=0, n-1, a(k)*x^k/k!); C=floor(exp(n))-sum(k=0, n-1, polcoeff(F^n+x*O(x^k), k, x)); G=sum(k=0, n-1, polcoeff(F^n+x*O(x^k), k, x)*x^k); A=n!*polcoeff((G+C*x^n)^(1/n)+x*O(x^n), n, x)); A}

Crossrefs

Cf. A100064.

Sequence in context: A225208 A290567 A339758 * A066807 A165497 A051752

Adjacent sequences: A100062 A100063 A100064 * A100066 A100067 A100068

Keyword

sign

Author

Paul D. Hanna, Nov 02 2004

Status

approved