%I #21 Oct 30 2023 09:48:45

%S 1,2,3,5,5,9,7,14,10,20,11,31,13,35,25,45,17,74,19,70,56,77,23,161,26,

%T 104,111,154,29,261,31,222,198,170,56,536,37,209,325,496,41,623,43,

%U 605,626,299,47,1407,50,602,731,1092,53,1305,517,1443,1026,464,59,4002,61,527,1429,2381,1352,2596,67,3009,1840,2787,71,6719,73,740,5378,4655,407,5135,79,10118

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

%H Seiichi Manyama, <a href="/A318636/b318636.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1024 from Paul D. Hanna)

%F a(n) = Sum_{d|n} binomial(n/d,d). - _Ridouane Oudra_, May 02 2019

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

%e G.f.: A(x) = x + 2*x^2 + 3*x^3 + 5*x^4 + 5*x^5 + 9*x^6 + 7*x^7 + 14*x^8 + 10*x^9 + 20*x^10 + 11*x^11 + 31*x^12 + 13*x^13 + 35*x^14 + 25*x^15 + 45*x^16 + ...

%e such that

%e A(x) = x + (1 + x^2)^2 - 1 + (1 + x^3)^3 - 1 + (1 + x^4)^4 - 1 + (1 + x^5)^5 - 1 + (1 + x^6)^6 - 1 + (1 + x^7)^7-1 + ...

%e RELATED SERIES.

%e The g.f. A(x) equals following series at y = 1:

%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 + 1 +x*O(x^n))^m - 1), n)}

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

%Y Cf. A143862, A318637, A318638.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Sep 07 2018