A318637 Expansion of Sum_{n>=1} ( (2 + x^n)^n - 2^n ).

1, 4, 12, 33, 80, 198, 448, 1048, 2305, 5200, 11264, 24824, 53248, 115360, 245800, 526081, 1114112, 2364064, 4980736, 10497290, 22020656, 46165504, 96468992, 201396028, 419430401, 872574976, 1811944704, 3758469400, 7784628224, 16107002892, 33285996544, 68721443936, 141733963008, 292062232576, 601295421524, 1236960724929, 2542620639232, 5222702645248, 10720238663680, 21990282376768

Offset: 1

Author

Paul D. Hanna, Sep 07 2018

Keywords

• nonn

Examples

			G.f.: A(x) = x + 4*x^2 + 12*x^3 + 33*x^4 + 80*x^5 + 198*x^6 + 448*x^7 + 1048*x^8 + 2305*x^9 + 5200*x^10 + 11264*x^11 + 24824*x^12 + 53248*x^13 + 115360*x^14 + ...
such that
A(x) = x + (2 + x^2)^2 - 2^2 + (2 + x^3)^3 - 2^3 + (2 + x^4)^4 - 2^4 + (2 + x^5)^5 - 2^5 + (2 + x^6)^6 - 2^6 + (2 + x^7)^7 - 2^7 + ...
RELATED SERIES.
The g.f. A(x) equals following series at y = 2:
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 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..1024

Crossrefs

Cf. A318636, A318638.

Programs

PARI

{a(n) = polcoeff( sum(m=1,n, (x^m + 2 +x*O(x^n))^m - 2^m), n)}
for(n=1,100, print1(a(n),", "))

PARI

a(n) = sumdiv(n, d, 2^(d-n/d)* binomial(d, n/d)); \\ Seiichi Manyama, Apr 24 2021

Formula

a(n) ~ n * 2^(n-1). - Vaclav Kotesovec, Oct 10 2020

a(n) = Sum_{d|n} 2^(d - n/d) * binomial(d, n/d). - Seiichi Manyama, Apr 24 2021

G.f.: Sum_{k >=1} x^(k^2)/(1-2*x^k)^(k+1). - Seiichi Manyama, Oct 30 2023