A157311 G.f.: exp( Sum_{n>=1} a(n)*x^n/n ) = Product_{n>=1} (1 + a(n-1)*x^n).

1, 1, 1, 4, 13, 66, 394, 2759, 22005, 198049, 1979646, 21776107, 261287398, 3396736175, 47553219799, 713298307974, 11412712029909, 194016104508454, 3492285524896921, 66353424973041500, 1327068107226627278, 27868430252187313730, 613105422439139763585

Offset

0,4

Links

Table of n, a(n) for n=0..22.

Formula

a(n) = Sum_{d divides n, 1<=d<=n} -d*(-a(d-1))^(n/d) for n>0 with a(0)=1.

Product_{n>=1} (1 + a(n-1)*x^n) = g.f. of A157312.

Example

Define G(x) by the exponential:
G(x) = exp(x + x^2/2 + 4*x^3/3 + 13*x^4/4 + 66*x^5/5 + 394*x^6/6 +...)
then G(x) also equals the product:
G(x) = (1 + x)(1 + x^2)(1 + x^3)(1 + 4*x^4)(1 + 13*x^5)(1 + 66*x^6)*...;
where the coefficients in both expressions are the same (with offset)
and G(x) is the g.f. of A157312:
G(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 18*x^5 + 84*x^6 + 481*x^7 + 3249*x^8 +...

Mathematica

a[0] = 1; a[n_] := a[n] = DivisorSum[n, -# * (-a[#-1])^(n/#) &]; Array[a, 20, 0] (* Amiram Eldar, Aug 15 2023 *)

Prog

(PARI) {a(n)=if(n==0, 1, sumdiv(n, d, if(d>=1&d<=n, -d*(-a(d-1))^(n/d))))}
(PARI) {a(n)=if(n==0, 1, n*polcoeff(1+sum(k=1, n, log(1+a(k-1)*x^k +x*O(x^n))), n))}
(PARI) {a(n)=if(n==0, 1, n*polcoeff(sum(k=1, n, -sum(j=1, n\k, (-a(k-1))^j*x^(k*j)/j)+x*O(x^n)), n))}

Crossrefs

Cf. A157312.

Sequence in context: A222771 A052415 A129433 * A318600 A309900 A096805

Adjacent sequences: A157308 A157309 A157310 * A157312 A157313 A157314

Keyword

nonn

Author

Paul D. Hanna, Mar 10 2009

Status

approved