%I #18 Oct 30 2023 09:52:41

%S 1,4,12,33,80,198,448,1048,2305,5200,11264,24824,53248,115360,245800,

%T 526081,1114112,2364064,4980736,10497290,22020656,46165504,96468992,

%U 201396028,419430401,872574976,1811944704,3758469400,7784628224,16107002892,33285996544,68721443936,141733963008,292062232576,601295421524,1236960724929,2542620639232,5222702645248,10720238663680,21990282376768

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

%H Paul D. Hanna, <a href="/A318637/b318637.txt">Table of n, a(n) for n = 1..1024</a>

%F a(n) ~ n * 2^(n-1). - _Vaclav Kotesovec_, Oct 10 2020

%F a(n) = Sum_{d|n} 2^(d - n/d) * binomial(d, n/d). - _Seiichi Manyama_, Apr 24 2021

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

%e 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 + ...

%e such that

%e 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 + ...

%e RELATED SERIES.

%e The g.f. A(x) equals following series at y = 2:

%e 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 + ...

%o (PARI) {a(n) = polcoeff( sum(m=1,n, (x^m + 2 +x*O(x^n))^m - 2^m), n)}

%o for(n=1,100, print1(a(n),", "))

%o (PARI) a(n) = sumdiv(n, d, 2^(d-n/d)* binomial(d, n/d)); \\ _Seiichi Manyama_, Apr 24 2021

%Y Cf. A318636, A318638.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Sep 07 2018