OFFSET
1,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..1024
FORMULA
a(n) ~ n * 3^(n-1). - Vaclav Kotesovec, Oct 10 2020
a(n) = Sum_{d|n} 3^(d - n/d) * binomial(d, n/d). - Seiichi Manyama, Apr 24 2021
G.f.: Sum_{k >=1} x^(k^2)/(1-3*x^k)^(k+1). - Seiichi Manyama, Oct 30 2023
EXAMPLE
G.f.: A(x) = x + 6*x^2 + 27*x^3 + 109*x^4 + 405*x^5 + 1467*x^6 + 5103*x^7 + 17550*x^8 + 59050*x^9 + 197100*x^10 + 649539*x^11 + 2126991*x^12 + ...
such that
A(x) = x + (3 + x^2)^2 - 3^2 + (3 + x^3)^3 - 3^3 + (3 + x^4)^4 - 3^4 + (3 + x^5)^5 - 3^5 + (3 + x^6)^6 - 3^6 + (3 + x^7)^7 - 3^7 + ...
RELATED SERIES.
The g.f. A(x) equals following series at y = 3:
Sum_{n>=1} ((y + x^n)^n - y^n) = x + 2*y*x^2 + 3*y^2*x^3 + (4*y^3 + 1)*x^4 + 5*y^4*x^5 + (6*y^5 + 3*y)*x^6 + 7*y^6*x^7 + (8*y^7 + 6*y^2)*x^8 + (9*y^8 + 1)*x^9 + (10*y^9 + 10*y^3)*x^10 + 11*y^10*x^11 + (12*y^11 + 15*y^4 + 4*y)*x^12 + 13*y^12*x^13 + (14*y^13 + 21*y^5)*x^14 + (15*y^14 + 10*y^2)*x^15 + (16*y^15 + 28*y^6 + 1)*x^16 + ...
PROG
(PARI) {a(n) = polcoeff( sum(m=1, n, (x^m + 3 +x*O(x^n))^m - 3^m), n)}
for(n=1, 100, print1(a(n), ", "))
(PARI) a(n) = sumdiv(n, d, 3^(d-n/d)* binomial(d, n/d)); \\ Seiichi Manyama, Apr 24 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 07 2018
STATUS
approved