A157313 G.f.: exp( Sum_{n>=1} a(n)*x^n/n ) = Product_{n>=1} 1/(1 - a(n-1)*x^n).

1, 1, 3, 10, 43, 216, 1326, 9283, 74667, 672085, 6730098, 74031079, 888657130, 11552542691, 161747905609, 2426218982400, 38820193151115, 659943283568956, 11879029341157575, 225701557481993926, 4514035666639844778, 94794749015757064732, 2085484976583065409751

Offset: 0

Author

Paul D. Hanna, Mar 10 2009

Keywords

• nonn

Examples

			Define G(x) by the exponential:
G(x) = exp(x + 3*x^2/2 + 10*x^3/3 + 43*x^4/4 + 216*x^5/5 + 1326*x^6/6 +...)
then 1/G(x) also equals the product:
1/G(x) = (1 - x)(1 - x^2)(1 - 3*x^3)(1 - 10*x^4)(1 - 43*x^5)(1 - 216*x^6)*...
where the coefficients in both expressions are the same (with offset)
and G(x) is the g.f. of A157314:
G(x) = 1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 62*x^5 + 298*x^6 + 1700*x^7 +...
		

Crossrefs

Cf. A157314, A157311.

Programs

Mathematica

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

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))}

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/(1 - a(n-1)*x^n) = g.f. of A157314.

Extensions

a(21)-a(22) from Amiram Eldar, Aug 15 2023