A322209 L.g.f.: log( Product_{n>=1} 1/(1 - (2^n+1)*x^n) ).

0, 3, 19, 54, 199, 408, 1612, 3090, 11023, 26487, 80994, 199686, 676540, 1700832, 5285096, 15197274, 45739039, 131368404, 401655943, 1172222958, 3549402474, 10533769146, 31617172980, 94336116834, 283990486780, 848323147233, 2546924693306, 7631598676410, 22903854049016, 68645946621360, 206035134959112, 617739968277066, 1853594327953471

Offset: 0

Author

Paul D. Hanna, Dec 01 2018

Keywords

• nonn

Examples

			L.g.f.: L(x) = 3*x + 19*x^2/2 + 54*x^3/3 + 199*x^4/4 + 408*x^5/5 + 1612*x^6/6 + 3090*x^7/7 + 11023*x^8/8 + 26487*x^9/9 + 80994*x^10/10 + 199686*x^11/11 + 676540*x^12/12 + ...
such that
exp( L(x) ) = 1 + 3*x + 14*x^2 + 51*x^3 + 195*x^4 + 663*x^5 + 2345*x^6 + 7707*x^7 + 25744*x^8 + 82980*x^9 + 267812*x^10 + 846150*x^11 + 2676163*x^12 + ... + A322199(n)*x^n + ...
also,
exp( L(x) ) = 1/( (1 - 3*x) * (1 - 5*x^2) * (1 - 9*x^3) * (1 - 17*x^4) * (1 - 33*x^5) * (1 - 65*x^6) * (1 - 129*x^7) * ... * (1 - (2^n+1)*x^n) * ... ).
		

Crossrefs

Cf. A322200, A322199.

Programs

PARI

{L = sum(n=1,41, -log(1 - (x^n + y^n) +O(x^41) +O(y^41)) );}
{A322200(n,k) = polcoeff( (n+k)*polcoeff( L,n,x),k,y)}
{a(n) = sum(k=0,n, A322200(n-k,k)*2^k )}
for(n=0,40, print1( a(n),", ") )

Formula

a(n) = Sum_{k=0..n} A322200(n-k,k) * 2^k for n >= 0.