A322199 Expansion of Product_{k>=1} 1/(1 - (2^k + 1) * x^k).

1, 3, 14, 51, 195, 663, 2345, 7707, 25744, 82980, 267812, 846150, 2676163, 8337189, 25947281, 80053128, 246468551, 754366239, 2305139065, 7014997404, 21317567297, 64606020012, 195557995054, 590855420007, 1783577678925, 5377112705874, 16199746640340, 48763788775530, 146712079122114, 441146762285301, 1326002750336702, 3984148679940612, 11967872331787643

Offset: 0

Author

Paul D. Hanna, Dec 01 2018

Keywords

• nonn

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1, g(n) = 2^n + 1. - Seiichi Manyama, Apr 11 2025

Examples

			G.f.: A(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 + ...
such that
A(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) * ... ).
RELATED SERIES.
log( A(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 + ... + A322209(n)*x^n/n + ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
• Seiichi Manyama, Generalized Euler transform.

Crossrefs

Cf. A322209, A322200, A322210.

Cf. A000051, A266964.

Programs

PARI

{a(n) = polcoeff( 1/prod(m=1,n, 1 - (2^m+1)*x^m +x*O(x^n)),n)}
for(n=0,30, print1(a(n),", "))

Formula

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} A322200(n-k,k) * 2^k ).

a(n) ~ c * 3^n, where c = Product_{k>=2} 1/(1 - (2^k + 1)/3^k) = 6.49344992975096517443610066284481821741772051973643441550853873760083... - Vaclav Kotesovec, Oct 04 2020

a(n) = Sum_{k=0..n} 2^k * A322210(k,n-k). - Seiichi Manyama, Apr 11 2025